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\begin{document}

\section{The Baum--Connes assembly map and the
conjecture}\label{the-baumconnes-assembly-map-and-the-conjecture}

\emph{Written by GPT-6.1 Sol (OpenAI). Public domain (CC0).}

Let \(G\) be second countable and locally compact. All spaces in the
construction are locally compact, Hausdorff and second countable, all
coefficient algebras are separable and graded, and all group actions
preserve grading. A proper action means that \((g,x)\mapsto(gx,x)\) is a
proper map. A \(G\)-compact space has compact orbit space. Haar measure
is left Haar measure, with modular function defined by \[
\int_G f(sg)\,ds=\Delta(g)^{-1}\int_G f(s)\,ds.
\tag{A1}
\] The convolution and involution conventions are those of
\href{KT-KK-18.html\#1-crossing-a-hilbert-module}{\emph{Descent and the
K-theory of crossed products}, Sections 1--3}. That lesson proves both
versions of descent, their naturality, their product compatibility and
the full-to-reduced compatibility square. The cutoff argument below
supplies the other factor in assembly.

\subsection{1. A normalized cutoff on a proper cocompact
space}\label{a-normalized-cutoff-on-a-proper-cocompact-space}

\textbf{Lemma A.1.} A proper \(G\)-compact space \(X\) has a nonnegative
\(c\in C_c(X)\) such that \[
\int_G c(s^{-1}x)^2\,ds=1\qquad(x\in X).
\tag{A2}
\] For compact \(K,L\subset X\), the transporter \[
T(K,L)=\{g\in G:gK\cap L\ne\varnothing\}
\tag{A3}
\] is compact.

\textbf{Proof.} The inverse image of \(L\times K\) under the proper
action map is compact. Its group projection is exactly \(T(K,L)\),
proving the last assertion. The orbit map \(X\to X/G\) is open: the
inverse image of the image of an open set is the union of its
translates. Around each point choose a precompact open neighbourhood
\(U\) and a nonnegative compactly supported continuous function positive
on \(U\). These functions exist by the proved cutoff construction in
\href{supporting/noncompact-foundations/noncompact-foundations.html\#ncf-001}{\emph{Noncompact
foundations for equivariant induction}, NCF.1}. Compactness of \(X/G\)
selects finitely many such \(U\)'s whose orbit images cover the
quotient. The sum \(b\) of their functions is positive somewhere on
every orbit. Its support \(K\) is compact, and \(GK=X\).

Set \[
h(x)=\int_G b(s^{-1}x)^2\,ds.
\tag{A4}
\] For \(x\) in a fixed compact neighbourhood \(L\), the integrand
vanishes unless \(s\in T(K,L)\). This is one compact integration set for
all such \(x\). Continuity and uniform continuity on that compact set
make \(h\) continuous there. Its value is positive: if
\(b(s_0^{-1}x)>0\), the same inequality holds on a nonempty open
neighbourhood of \(s_0\), which has positive Haar measure by
\href{supporting/noncompact-foundations/noncompact-foundations.html\#ncf-004}{NCF.4}.
Substitution \(s=gt\), using left invariance, gives \(h(gx)=h(x)\). Thus
\(c=b h^{-1/2}\) is continuous, nonnegative and compactly supported, and
its orbit integral is \(h(x)/h(x)=1\). All integrals here have compact
support locally in \(x\); the proved integration results in
\href{supporting/noncompact-foundations/noncompact-foundations.html\#ncf-005}{NCF.5}
therefore apply. \(\square\)

Write \(D_X=C_0(X)\rtimes G\) and \(D_{X,r}=C_0(X)\rtimes_rG\). Both
contain the same compactly supported convolution algebra, with action
\((\alpha_g a)(x)=a(g^{-1}x)\).

\textbf{Proposition A.2 (the cutoff projection).} The function \[
p_c(g,x)=\Delta(g)^{-1/2}c(x)c(g^{-1}x)
\tag{A5}
\] is a projection in \(D_X\), and its image is a projection in
\(D_{X,r}\). Its K-theory class is independent of \(c\).

\textbf{Proof.} A nonzero value requires \(x\in\operatorname{supp}c\)
and \(g^{-1}x\in\operatorname{supp}c\). Lemma A.1 bounds \(g\) in a
compact transporter. Thus \(p_c\in C_c(G\times X)\). The involution is
\[
f^*(g,x)=\Delta(g)^{-1}\overline{f(g^{-1},g^{-1}x)}.
\tag{A6}
\] Inserting (A5) gives \(p_c^*=p_c\). Convolution gives \[
\begin{aligned}
(p_c*p_c)(g,x)
&=\int_G p_c(s,x)p_c(s^{-1}g,s^{-1}x)\,ds\\
&=\Delta(g)^{-1/2}c(x)c(g^{-1}x)
       \int_G c(s^{-1}x)^2\,ds\\
&=p_c(g,x).
\end{aligned}
\tag{A7}
\] The identities hold before either completion. In particular, the
modular factor in (A5) is needed for self-adjointness when \(G\) is not
unimodular.

If \(c_0,c_1\) satisfy (A2), so does \[
c_t(x)=\sqrt{(1-t)c_0(x)^2+t c_1(x)^2}\quad(0\leq t\leq1).
\tag{A8}
\] This is a continuous path in the uniform norm, supported in one
compact set. Its projections have one compact group support by (A3);
their coefficients converge uniformly there. The crossed-product norm is
bounded by the group \(L^1\) coefficient norm, so \(t\mapsto p_{c_t}\)
is norm continuous in both completions. It is a projection homotopy,
proving independence.

There is also an explicit comparison useful later. For normalized
\(c,d\), put \[
v_{c,d}(g,x)=\Delta(g)^{-1/2}c(x)d(g^{-1}x).
\tag{A9}
\] The same multiplication gives \(v_{c,d}^*=v_{d,c}\) and
\(v_{c,d}v_{d,e}=v_{c,e}\). Hence \(v_{c,d}^*v_{c,d}=p_d\) and
\(v_{c,d}v_{c,d}^*=p_c\). This is an actual partial isometry in either
crossed product. \(\square\)

We write \([p_X]\) for the resulting even class. As a scalar-source
KK-cycle it is \((p_cD_{X,(r)},\mathbb C,0)\): the identity of this
singly generated module is the compact rank-one operator
\(\theta_{p_c,p_c}\), so the zero operator has the required compact
defect. The partial isometry (A9) or the projection homotopy (A8) gives
the same KK-class for another cutoff. The full-to-reduced quotient sends
the full class to the reduced class; no equality of the two
crossed-product algebras is required for this assertion.

\subsection{2. Naturality in spaces and
coefficients}\label{naturality-in-spaces-and-coefficients}

\textbf{Lemma A.3.} Every continuous equivariant map \(f:X\to Y\)
between proper \(G\)-compact spaces is proper. Pullback carries the
cutoff class of \(Y\) to that of \(X\): \[
[p_Y]\widehat\otimes_{D_{Y,(r)}}[f^*\rtimes_{(r)}G]=[p_X].
\tag{A10}
\]

\textbf{Proof.} By Lemma A.1 choose a compact \(K\subset X\) with
\(GK=X\). For compact \(L\subset Y\), write \(x=gk\in f^{-1}(L)\). Then
\(g f(k)\in L\), so \(g\in T(f(K),L)\). Consequently \(f^{-1}(L)\) is a
closed subset of the compact set \(T(f(K),L)K\), and is compact.
Pullback is therefore a homomorphism \(f^*:C_0(Y)\to C_0(X)\). It is
nondegenerate, since a function equal to one on \(f(K')\), for compact
\(K'\subset X\), acts as one on every function supported in \(K'\).

If \(c_Y\) is a normalized cutoff, \(c_Y\circ f\) is compactly supported
by properness and is normalized by equivariance. Pullback of (A5) is
exactly its projection. The descended homomorphism class is the class of
the crossed homomorphism, by equation (2.6) of the descent lesson.
Proposition A.2 gives (A10). \(\square\)

For a separable \(G\)-algebra \(A\), define \[
\begin{aligned}
\mu_{X,A,(r)}:KK_i^G(C_0(X),A)&\longrightarrow K_i(A\rtimes_{(r)}G),\\
\mu_{X,A,(r)}(x)&=[p_X]\widehat\otimes_{D_{X,(r)}}j_{G,(r)}(x).
\end{aligned}
\tag{A11}
\] For graded coefficients use the scalar-cycle notation
\(K_i(B)=KK_i(\mathbb C,B)\). Thus (A11) is a construction entirely in
the proved KK category. The comparison with projection and unitary
K-theory for trivially graded coefficients is a separate coefficient
Fredholm-picture result.

\textbf{Theorem A.4.} Formula (A11) is a well-defined group
homomorphism, independent of the cutoff. For an equivariant map
\(f:X\to Y\), put \(f_*x=[f^*]\widehat\otimes_{C_0(X)}x\). Then \[
\mu_{Y,A,(r)}(f_*x)=\mu_{X,A,(r)}(x).
\tag{A12}
\] For \(\eta\in KK_j^G(A,B)\), it satisfies \[
\mu_{X,B,(r)}(x\widehat\otimes_A\eta)
=\mu_{X,A,(r)}(x)\widehat\otimes_{A\rtimes_{(r)}G}j_{G,(r)}(\eta).
\tag{A13}
\] The reduced map is the full map followed by the coefficient quotient:
\[
\mu_{X,A,r}(x)=\mu_{X,A}(x)\widehat\otimes_{A\rtimes G}[q_A].
\tag{A14}
\]

\textbf{Proof.} Proposition A.2 gives a cutoff-independent class.
Descent respects cycle homotopy and addition by Theorem 2.2 of the
descent lesson, and product by its Theorem 3.2. Associativity of the
proved product therefore makes (A11) a well-defined homomorphism. For
(A12), product compatibility gives

\([p_Y]j(f_*x)=[p_Y][f^*\rtimes G]j(x)=[p_X]j(x)\),

with the same calculation in the reduced completion. This uses Lemma A.3
at its middle step. For (A13), expand \(j(x\eta)=j(x)j(\eta)\) and
reassociate; the cutoff factor is even, so no sign arises. Finally (3.9)
of the descent lesson says \([q_{C_0(X)}]j_{G,r}(x)=j_G(x)[q_A]\).
Multiply on the left by the full cutoff class, whose quotient is the
reduced cutoff class, to obtain (A14). \(\square\)

If \(\mathcal E\) is a given model for the classifying space of proper
actions, take the directed family of its closed, locally compact,
\(G\)-compact subspaces used to define equivariant K-homology with
\(G\)-compact supports. With coefficients the definition is \[
RK_i^G(\mathcal E;A)=\underset{X\subset\mathcal E}{\operatorname{colim}}\,
KK_i^G(C_0(X),A).
\tag{A15}
\] The structure map for \(X\subset Y\) is pullback of the source
representation along restriction \(C_0(Y)\to C_0(X)\). Formula (A12)
says exactly that (A11) is constant under these structure maps. It
therefore induces a unique assembly homomorphism \[
\mu_{A,r}:RK_i^G(\mathcal E;A)\longrightarrow K_i(A\rtimes_rG).
\tag{A16}
\] This step is purely a passage to a direct limit of the proved maps.
Construction of the classifying model and the independence of its choice
are separate geometric inputs; none is inferred from a cutoff
calculation. For an equivariant homotopy through proper maps of
\(G\)-compact spaces, the corresponding source homomorphisms form a
homotopy, so (A12) respects that homotopy as well.

\subsection{3. The finite-group
calculation}\label{the-finite-group-calculation}

For finite \(G\), use Haar probability measure. The point is a
classifying proper space: every subgroup has the point as a contractible
fixed set, and there is exactly one equivariant map from any space to
it. We may take \(X=\{*\}\), \(c=1\). In counting-measure notation its
projection is \[
p_G=\frac1{|G|}\sum_{g\in G}u_g.
\tag{A17}
\]

\textbf{Theorem A.5.} For every separable graded finite-group algebra
\(A\), (A16) is the Green--Julg isomorphism \[
KK_i^G(\mathbb C,A)\ \cong\ K_i(A\rtimes_rG).
\tag{A18}
\]

\textbf{Proof.} We identify the assembly cycle with the cycle
isomorphism already proved in
\href{KT-KK-17.html\#4-invariant-compact-operators-and-green-julg}{\emph{Equivariant
KK-theory and the Green--Julg theorem}, Theorem 4.3}. That theorem
proves all homotopy relations as well as the bijection. Its
normalization makes the scalar representation unital and the operator
invariant, adds a degenerate balanced regular module and applies
equivariant stabilization. Thus it suffices to compare cycles on finite
sums and the countable balanced amplification of \[
\mathcal R_A=L^2(G)\otimes A,\qquad
(U_s\xi)(t)=\alpha_s(\xi(ts)).
\tag{A19}
\] Let \(D=A\rtimes G\). Sections of its crossed module have variables
\(h,t\in G\). The descended scalar group representation is \[
(V_s\zeta)(h)(t)=\alpha_s(\zeta(s^{-1}h)(ts)).
\tag{A20}
\] The projection (A17) acts as the average of these \(V_s\)'s. Its
range is exactly their invariant submodule.

For \(f\in C(G,A)\), define \[
\zeta_f(h)(t)=\alpha_{t^{-1}}(f(th)).
\tag{A21}
\] It is invariant by substitution into (A20). Conversely, for invariant
\(\zeta\), use \(s=t^{-1}\) in (A20) to obtain (A21) with
\(f(r)=\zeta(r)(e)\). Since \(G\) is finite, these are bounded
operations on finite tuples, with no point evaluation on an arbitrary
\(L^2\)-function.

Using probability measure in both variables, equation (1.2) of the
descent lesson gives \[
\begin{aligned}
\langle\zeta_f,\zeta_k\rangle_D(r)
&=\int_G\int_G
 \alpha_{h^{-1}}\alpha_{t^{-1}}
      (f(th)^*k(thr))\,dt\,dh\\
&=\int_G\alpha_{s^{-1}}(f(s)^*k(sr))\,ds\\
&=(f^**k)(r).
\end{aligned}
\tag{A22}
\] For the second equality set \(s=th\) for each \(t\); the remaining
\(t\)-integral has mass one. Right convolution in the crossed module
similarly gives \(\zeta_f b=\zeta_{f*b}\). Thus \(f\mapsto\zeta_f\)
extends to a unitary \[
D\ \longrightarrow\ p_G(\mathcal R_A\rtimes G).
\tag{A23}
\] Surjectivity follows from the converse to (A21), first on finite
tuples and then on their completion. Amplifying gives the unitary for
the balanced standard \(D\)-module.

The invariant compact algebra on \(\mathcal R_A\) is \(D\) by
Proposition 4.1 of the equivariant lesson. Its kernel for
\(a\in C(G,A)\) is the integrated covariant operator \[
(K_a\xi)(t)=\int_G\alpha_{t^{-1}}(a(s))\xi(s^{-1}t)\,ds.
\tag{A24}
\] Substitution of (A21) gives \(K_a\zeta_f=\zeta_{a*f}\). Hence (A23)
identifies the descended invariant compact operators with left
multiplication by \(D\). It identifies their invariant multipliers as
well: multiply by a compact-algebra approximate identity, use (A24), and
pass to the strict limit on the generated module. This is precisely the
multiplier extension proved in Lemma 4.2 of that lesson.

An invariant normalized cycle operator \(F\) commutes with \(p_G\). Its
assembly product is therefore its restriction to the range of \(p_G\);
compression preserves its compact defects. Under (A23), this is the same
odd multiplier of the standard compact algebra used by the Green--Julg
cycle comparison. Degenerate cycles and the balancing added in
normalization map to their corresponding degenerate cycles. Thus the two
homomorphisms agree on every class, and Theorem 4.3 proves that assembly
is an isomorphism. Full and reduced crossed products agree for compact
groups by the faithful regular-representation proof in Proposition 4.1,
so the conclusion is the reduced one. A trivial right Clifford factor
passes unchanged through (A19)--(A24), proving both degrees. \(\square\)

\subsection{4. A concrete lattice cutoff and the trace
implication}\label{a-concrete-lattice-cutoff-and-the-trace-implication}

For \(\mathbb Z\) acting on \(\mathbb R\) by translations, let \[
h(x)=\max(1-|x|,0),\qquad c(x)=\sqrt{h(x)}.
\tag{A25}
\] If \(x=k+t\), \(k\in\mathbb Z\), \(0\leq t\leq1\), the only possibly
nonzero translates are \(h(x-k)=1-t\) and \(h(x-k-1)=t\). Their sum is
one, including endpoints. Thus (A2) holds and \[
p(n,x)=\sqrt{h(x)h(x-n)}.
\tag{A26}
\] It vanishes for \(|n|\geq2\), and at \(|n|=1\) has support in the
overlap of adjacent intervals. Formula (A7) proves that it is a
projection; its class is the concrete cutoff class, independent of the
tent-shaped choice. For \(\mathbb Z^d\) acting on \(\mathbb R^d\), use
\(\prod_j h(x_j)\); the orbit sum factors into \(d\) sums equal to one.
This computes the cutoff projection, rather than yet asserting the
lattice assembly isomorphism.

Here is the precise elementary implication of trace integrality. For a
countable discrete group \(\Gamma\), the reduced regular representation
defines \[
\tau(a)=\langle\delta_e,a\delta_e\rangle
\quad(a\in C_r^*(\Gamma)).
\tag{A27}
\] On finite group-algebra sums it is the coefficient of \(e\). In a
product the contributions to that coefficient are paired by
\(g\leftrightarrow g^{-1}\), so \(\tau(ab)=\tau(ba)\). Density and
continuity extend this identity. Also
\(\tau(a^*a)=\|a\delta_e\|^2\geq0\) and \(\tau(1)=1\). Right
translations commute with the reduced algebra. If \(\tau(a^*a)=0\), then
\(a\delta_e=0\) and consequently \(a\delta_g=0\) for every \(g\); hence
\(a=0\). The trace is faithful.

For projections \(p\in M_n(C_r^*(\Gamma))\), put
\(\tau_n(p)=\sum_j\tau(p_{jj})\). A rectangular partial isometry \(v\)
satisfies \(\tau_n(vv^*)=\tau_m(v^*v)\), by expanding the matrix entries
and using the trace identity. Adding a zero block does not change this
value, and block sums add it. It therefore extends to a homomorphism
\(\tau_*:K_0(C_r^*(\Gamma))\to\mathbb R\), using the projection
definition of K-theory proved in the index-pairing lesson.

\textbf{Proposition A.6.} If
\(\tau_*(K_0(C_r^*(\Gamma)))\subset\mathbb Z\), the only projections and
the only idempotents in \(C_r^*(\Gamma)\) are \(0,1\). More generally,
if assembly is surjective and its image has integral trace, this
conclusion holds.

\textbf{Proof.} A projection \(p\) has \(0\leq\tau(p)\leq1\).
Integrality gives value zero or one. Faithfulness applied to \(p^*p=p\),
or to \((1-p)^*(1-p)=1-p\), gives \(p=0\) or \(p=1\).

For an idempotent \(e\), put \(H=1-(e-e^*)^2\geq1\). Expansion shows
that \(H\) commutes with \(e,e^*\) and that \(He=ee^*e\). Consequently
\(p=ee^*H^{-1}\) is self-adjoint and \[
p^2=p,\qquad pe=e,\qquad ep=p.
\tag{A28}
\] For example \(H ee^*=ee^*ee^*\) proves the first identity. Thus
\(p=0\) forces \(e=0\), while \(p=1\) and \(ep=p\) force \(e=1\). The
surjectivity variant follows because every K-class then belongs to the
integral-trace image. \(\square\)

Theorem A.10 below proves integral trace on the entire torsion-free
assembly image. Combined with this proposition, it gives the projection
and idempotent consequence under surjectivity.

\subsection{5. Covering Dirac classes and the Mishchenko
index}\label{covering-dirac-classes-and-the-mishchenko-index}

Let \(\Gamma\) be a countable discrete group acting freely and properly
by spin\(^{c}\) isometries on a connected manifold \(X\), with closed
quotient \(M=X/\Gamma\). The spin\(^{c}\) structure and connection on
\(X\) are the lifts of those on \(M\). A lifted finite-rank Hermitian
coefficient bundle is allowed. Put \(A=C^*(\Gamma)\), or
\(A=C_r^*(\Gamma)\), and denote its canonical unitaries by \(u_\gamma\).
The algebra \(A\) is unital and separable. The Mishchenko bundle is \[
\mathcal L=X\times_\Gamma A,\qquad
\gamma(x,a)=(\gamma x,u_\gamma a).
\tag{M1}
\] Its transition maps are left multiplication, hence are right
\(A\)-linear unitaries. This convention matters.

Clifford multiplication is self-adjoint, with \(c(v)^2=|v|^2\); the
spinor Dirac convention is \(D=-i\sum_jc(e_j)\nabla_{e_j}\). In even
dimension use its spinor grading. In odd dimension use the positive
spinor representation followed by the last right \(Cl_1\) factor.

\subsubsection{5.1. Smooth cutoffs and the lifted
operator}\label{smooth-cutoffs-and-the-lifted-operator}

\textbf{Lemma M.1.} There is a smooth compactly supported real cutoff
\(c\geq0\) with \[
\sum_{\gamma\in\Gamma}c(\gamma^{-1}x)^2=1.
\tag{M2}
\] The lifted spinor Dirac \(D_X\), initially on smooth compactly
supported sections, is self-adjoint. It has locally compact resolvents
and represents an equivariant cycle with source \(C_0(X)\).

\textbf{Proof.} Freeness and properness give covering charts. Indeed,
choose a compact neighbourhood of \(x\). Only finitely many of its
translates can meet it, by the compact-transporter proof of Lemma A.1.
For each nonidentity element in that finite set choose a smaller
neighbourhood of \(x\) disjoint from its translate, using
\(\gamma x\ne x\). Their finite intersection has disjoint translates.
Its quotient image is a chart, and the quotient map restricts to a
diffeomorphism there.

Choose finitely many relatively compact such charts whose quotient
images cover \(M\), with smooth nonnegative bump functions positive on
smaller charts. Euclidean bump functions and a finite partition produce
them. Let \(b\) be their sum on \(X\). The sum
\(h(x)=\sum_\gamma b(\gamma^{-1}x)^2\) is locally finite, smooth,
positive and invariant: local finiteness follows from the compact
transporter of \(\operatorname{supp}b\) with any compact neighbourhood.
Then \(c=b/\sqrt h\) proves (M2). Every derivative of \(c\) is bounded,
since its support is compact.

Here is a proper smooth exhaustion with bounded gradient, without
assuming a compact base argument on \(X\). Let
\(K=\operatorname{supp}c\), \(U=\{c>0\}\), and
\(S=\{\gamma:\gamma K\cap K\ne\varnothing\}\). This is finite and
symmetric. The translates of the open set \(U\) cover connected \(X\).
Their intersection graph is connected; otherwise the unions over its two
collections of components would partition \(X\) into disjoint nonempty
open sets. Edges from \(\gamma U\) to \(\delta U\) have
\(\gamma^{-1}\delta\in S\). Hence \(S\) generates \(\Gamma\). Let
\(\ell\) be its word length and set \[
r(x)=\sum_\gamma \ell(\gamma)c(\gamma^{-1}x)^2.
\tag{M3}
\] The sum is locally finite. If \(x\in\gamma_0K\), all active indices
belong to \(\gamma_0S\). Subtract \(\ell(\gamma_0)\) when
differentiating, using (M2). The inequalities
\(\lvert\ell(\gamma)-\ell(\gamma_0)\rvert\leq\max_{s\in S}\ell(s)\), the
finite size of \(S\), and the common gradient bound on translates of
\(c^2\) give a uniform bound on \(\lvert dr\rvert\). If \(r(x)\leq R\),
one active index has word length at most \(R\); thus \(x\) belongs to a
finite union of translates of \(K\). The sublevel is closed, hence
compact.

We give the domain details needed below. The exact elliptic parametrix
and Sobolev estimate in
\href{KT-KK-11.html\#4-spectral-triples-and-the-circle}{\emph{Unbounded
Kasparov modules and spectral triples}, Lemmas 4.2a--4.2b} apply on any
selected compact coordinate support: their kernels are cut off in that
support and their proof is local. In particular a distributional
solution of \(D_Xv\in L^2_{\mathrm{loc}}\) belongs to
\(H^1_{\mathrm{loc}}\). To see the estimate explicitly, the
order-minus-one local parametrix applied to \(D_Xv\), plus its smoothing
error and a commutator supported in the larger chart, bounds the \(H^1\)
norm on the smaller support by the local \(L^2\) norms of \(v,D_Xv\). A
finite partition gives the estimate on every compact set. Smooth
approximation in \(H^1\) therefore approximates compactly supported such
vectors in the graph norm, since the coefficients and their first
derivatives are bounded on that support.

The lifted Dirac is formally symmetric by integration by parts and
compatibility of its Clifford connection. Explicitly, the extra adjoint
term in an orthonormal frame is
\(-i\sum_jc(\nabla_{e_j}e_j+(\operatorname{div}e_j)e_j)\). Its
coefficient along \(e_k\) is zero because metric compatibility gives
\(\operatorname{div}e_k=-\sum_j\langle\nabla_{e_j}e_j,e_k\rangle\). If
\(D_X^*v=\pm iv\), the preceding local regularity applies. Choose
\(\chi_R=\chi(r/R)\), with \(\chi=1\) on \([0,1]\), zero on
\([2,\infty)\), and bounded derivative. The commutator is Clifford
multiplication by \(d\chi_R\), of norm at most \(C/R\). Formal symmetry,
justified by the compactly supported graph approximation, gives \[
\|\chi_Rv\|^2
\leq \|[D_X,\chi_R]v\|\,\|\chi_Rv\|
\leq C R^{-1}\|v\|\,\|\chi_Rv\|.
\tag{M4}
\] Since \(\chi_Rv\to v\), both deficiency spaces vanish. For clarity,
this proves self-adjointness: the ranges of the closed minimal
\(D_X\pm i\) are closed by \(\|(D_X\pm i)w\|^2=\|D_Xw\|^2+\|w\|^2\);
their orthogonal complements are those deficiency spaces, so both ranges
are all of \(L^2\). Smooth compactly supported sections remain a graph
core by definition of that minimal closure.

For \(a\in C_c^\infty(X)\), the local estimate sends
\(a(D_X\pm i)^{-1}\) into a bounded \(H^1\) set on a compact support.
This inclusion in \(L^2\) is compact: on each chart high Fourier
frequencies have norm bounded by \(R^{-1}\) times the \(H^1\) bound, and
the low-frequency cutoff has a square-integrable kernel on the two
compact supports, approximable by finite sums of product functions. A
finite partition proves compactness. Approximation extends it to
\(a\in C_0(X)\). Smooth compactly supported functions preserve the
domain and have bounded Clifford commutators. The proved
bounded-transform theorem,
\href{KT-KK-11.html\#2-the-bounded-transform}{Theorem 2.2 of that
lesson}, now gives the cycle. The group preserves the lifted connection
and its core, so it commutes with both resolvents and the transform. For
odd dimension use the fixed right \(Cl_1\) transfer throughout.
\(\square\)

\subsubsection{5.2. The range of the assembly
projection}\label{the-range-of-the-assembly-projection}

Write \(\mathcal H=L^2(X,S_X)\), including any lifted finite-rank
coefficient bundle. Descent identifies its coefficient module with
\(\mathcal H\otimes A\). On finite sections the identification sends
\(\zeta:\Gamma\to\mathcal H\) to \(\sum_h\zeta(h)\otimes u_h\); indeed,
\[
\left\langle\sum_h\zeta(h)\otimes u_h,
                  \sum_k\eta(k)\otimes u_k\right\rangle
=\sum_r\left(\sum_h\langle\zeta(h),\eta(hr)\rangle\right)u_r,
\tag{M5}
\] which is exactly the descended inner product. The right convolution
agrees as well, and these finite sections are dense. A crossed element
\(a u_\gamma\) acts by \[
(\phi(a)U_\gamma)\otimes L_{u_\gamma}.
\tag{M6}
\] The descended transform is \(F_{D_X}\otimes1\). These statements hold
for either completion by the actual descent construction in
\href{KT-KK-18.html\#1-crossing-a-hilbert-module}{\emph{Descent and the
K-theory of crossed products}, Section 1}.

The cutoff projection acts on this module as \[
P=\sum_\gamma
 \phi\!\left(c\,(c\circ\gamma^{-1})\right)
                U_\gamma\otimes L_{u_\gamma}.
\tag{M7}
\] Only finitely many terms are nonzero, by the transporter of \(K\)
with itself. Lemma A.2 proves that \(P\) is a self-adjoint projection.

Let \(\mathcal H_{\mathcal L}=L^2(M,S_M\otimes\mathcal L)\), with its
\(A\)-valued inner product given by integration over \(M\). A section
lifts to an \(A\)-valued section \(\Psi\) on \(X\) satisfying \[
\Psi(\gamma x)=(\gamma_S\otimes L_{u_\gamma})\Psi(x).
\tag{M8}
\] For a smooth section with finite group-algebra coefficients in
finitely many quotient charts define \(J\Psi=c\Psi\). Such sections are
dense: finite trivializing charts and a partition reduce approximation
to scalar smooth functions and to the dense group algebra in \(A\). On
the compact support of \(c\), their lifted coefficients use finitely
many deck transitions. Thus \(J\Psi\) belongs to the algebraic dense
part of \(\mathcal H\otimes A\).

\textbf{Lemma M.2.} Multiplication by \(c\) extends to an even unitary
\[
J:\mathcal H_{\mathcal L}\longrightarrow P(\mathcal H\otimes A).
\tag{M9}
\]

\textbf{Proof.} The pointwise \(A\)-valued inner product of two lifts in
(M8) is invariant, since both the spinor action and left multiplication
by \(u_\gamma\) are unitary. On a quotient chart, integrate over its
disjoint lifts. Equation (M2) says that the weights \(c^2\) sum to one.
A finite partition on \(M\) gives \[
\int_X c(x)^2\langle\Psi(x),\Phi(x)\rangle\,dx
 =\int_M\langle\Psi,\Phi\rangle\,dy.
\tag{M10}
\] This proves isometry with the exact \(A\)-valued inner product, not
merely after applying a trace. Inserting (M8) into (M7) gives
\(PJ\Psi=J\Psi\).

For a compactly supported smooth \(Z\) with finite \(A\)-coefficients on
\(X\), define \[
\Psi_Z(x)=\sum_\gamma c(\gamma^{-1}x)
       (\gamma_S\otimes L_{u_\gamma})Z(\gamma^{-1}x).
\tag{M11}
\] This is locally finite and smooth; on each compact quotient chart
only finitely many terms contribute, by properness and the support of
\(Z\). Changing \(\gamma\) to \(s\gamma\) proves (M8). Formula (M7)
gives \(J\Psi_Z=PZ\). Such \(Z\)'s are dense, so their projected vectors
are dense in the range of \(P\). The isometry has closed range, proving
surjectivity. \(\square\)

\subsubsection{5.3. Domains, product recognition and the twisted
Dirac}\label{domains-product-recognition-and-the-twisted-dirac}

Put \(D_0=D_X\otimes1\) on \(\mathcal H\otimes A\). It is self-adjoint
regular by the tensor-resolvent proof of
\href{KT-KK-11.html\#0-regular-calculus-localization-and-compact-supports}{Lemma
0.0 in the unbounded lesson}: the two resolvents are
\((D_X\pm i)^{-1}\otimes1\), have dense ranges, and satisfy the inverse
and adjoint identities. Its graph core consists of smooth compactly
supported spinors with finite group-algebra coefficients.

The finite sum (M7) preserves that core and has bounded commutator with
\(D_0\). The group terms commute with \(D_X\); the remaining commutators
are Clifford multiplications by derivatives of the finitely many smooth
compactly supported coefficients in (M7). Also \(P(D_0\pm i)^{-1}\) is
compact: each term is a localized compact Hilbert-space resolvent
tensored with left multiplication by a unitary in the unital algebra
\(A\). A finite-rank Hilbert-space approximation tensored with that
unitary is a finite-rank Hilbert-\(A\) operator.

\textbf{Lemma M.3.} The operator \[
T=P D_0 P\quad\hbox{on }P(\mathcal H\otimes A)
\tag{M12}
\] is self-adjoint regular, has compact resolvents, and represents the
product of the cutoff class with the descended Dirac cycle.

\textbf{Proof.} Remove the off-diagonal blocks of \(D_0\) relative to
\(P\). Their boundedness is precisely the bounded commutator just
proved. The resulting block-diagonal operator \(D_{\mathrm{diag}}\) is a
bounded self-adjoint perturbation of \(D_0\). At sufficiently large
imaginary parameter its inverse is the convergent Neumann series built
from the inverse of \(D_0\); the adjoint inverse has the opposite
parameter. These inverses prove self-adjoint regularity and preserve
both projection ranges. Restricting them to the first range proves that
assertion for \(T\), with domain \(P\operatorname{Dom}D_0\).

The resolvent identity expresses \(P(D_{\mathrm{diag}}-z)^{-1}\) as
\(P(D_0-z)^{-1}\) plus that same compact first factor times bounded
operators. Hence the compressed resolvent is compact. The projected core
remains a graph core, since projection is bounded for the graph norm by
the bounded commutator.

The first product module is \(p(C_0(X)\rtimes_{(r)}\Gamma)\), with
operator zero. Its smooth finite crossed-algebra columns, multiplied by
\(p\), are dense. For such a column \(\xi\), its represented commutator
with \(D_0\) is bounded by the same finite derivative calculation as
above. This gives the creation-operator error between \(T\) and \(D_0\);
taking adjoints gives the adjoint error. The commutator identity first
holds on the core, and bounded extension gives both domain inclusions,
including the adjoint inclusion via the self-adjoint maximal domain.
Positivity and domain conditions relative to the first operator zero are
automatic. The actual product criterion,
\href{KT-KK-11.html\#5-recognizing-unbounded-products}{Theorem 5.4 of
the unbounded lesson}, therefore identifies the bounded transform of
\(T\) with \([p]\otimes j_{\Gamma,(r)}[D_X]\). \(\square\)

The flat connection on \(\mathcal L\) is differentiation in each
covering trivialization; its transition maps \(u_\gamma\) are constant.
The twisted Dirac \(D_{\mathcal L}\) on \(M\) is consequently the
operator whose lift is \(D_X\) acting on the spinor part of (M8).

On the dense smooth sections used above, the Leibniz rule and (M7) give
\[
\begin{aligned}
PD_0J\Psi
&=c D_X\Psi
   -ic\,\mathrm{cl}\!\left(
       \sum_\gamma(c\circ\gamma^{-1})\,d(c\circ\gamma^{-1})
                     \right)\Psi\\
&=cD_X\Psi=JD_{\mathcal L}\Psi.
\end{aligned}
\tag{M13}
\] The middle sum is half the derivative of (M2), hence zero. This
identity includes the cutoff derivative term; it has not been discarded
as a compact error.

The inverse in (M11) carries the projected smooth core to smooth twisted
sections. Conversely the finite local-coefficient smooth sections are
mapped by \(J\) to that projected core. Lemma M.3 thus identifies the
closures and their domains: \[
J^{-1}TJ=\overline{D_{\mathcal L}}.
\tag{M14}
\] In particular the twisted Dirac is self-adjoint regular with compact
resolvents, without appealing to an unproved Hilbert-module elliptic
index theorem.

\textbf{Theorem M.4.} For the lifted Dirac class of the free cocompact
covering, \[
\mu_{X,(r)}([D_X])=\operatorname{Ind}_A(D_{\mathcal L}).
\tag{M15}
\] The right side is the scalar KK class of its bounded transform,
identified with \(K_{\dim M\bmod2}(A)\) by the proved scalar Fredholm
pictures in
\href{KT-KK-07.html\#2-the-even-scalar-fredholm-picture}{\emph{Pictures
of KK}, Sections 2--3 and 9}. All statements apply to full and reduced
group algebras, and to a lifted finite-rank coefficient twist.

\textbf{Proof.} The definition of assembly is the product in Lemma M.3.
Equations (M9) and (M14) identify its cycle with the twisted Dirac
cycle. In odd dimension every unitary and projection above is even and
commutes with the last right Clifford factor; the calculation therefore
passes through the exact odd convention unchanged. The full-to-reduced
map carries (M7), the inner products and the unitary to their reduced
versions. Its effect is the coefficient change already proved for
descent, so it carries the full index to the reduced index. \(\square\)

\subsubsection{5.4. Removing connectedness of the
covering}\label{removing-connectedness-of-the-covering}

The covering comparison (M15) also holds for any countable principal
free cocompact \(\Gamma\)-covering of a closed spin\(^{c}\) manifold,
including a disconnected total space. This extension preserves the
preceding connected-cover argument as its proved local input.

\textbf{Proof.} First take one connected component of the base and one
component \(X_0\) of its covering. The stabilizer \(\Gamma_0\) of
\(X_0\) acts freely and properly on it, and \(X_0/\Gamma_0\) is that
base component: any two lifts of the same point differ by a unique deck
element, which must preserve the selected component if both lifts lie
there. The other components are precisely the translates indexed by
\(\Gamma/\Gamma_0\). The connected-cover proof M.1 therefore proves
essential self-adjointness and localized compact resolvents for its
Dirac. On the entire covering the Dirac is the orthogonal direct sum of
those unitarily equivalent component operators. Its self-adjoint domain
consists of the vectors with square-summable component graph norms; the
finite-component smooth compactly supported cores form a graph core.
This follows directly by first truncating the graph-norm sum and then
using the component cores. Both opposite resolvents are the bounded
componentwise resolvents, so the direct sum is self-adjoint.

Every compact subset of the manifold meets finitely many components,
since its open component cover has a finite subcover. A compactly
supported multiplier therefore involves only finitely many of the
compact localized resolvents; it gives a compact operator on the direct
sum. The group action and the Dirac intertwine on components. A closed
manifold has finitely many base components by the same argument, so
summing over them causes no further issue. Countability follows from
second countability of the covering and makes the Hilbert space
separable.

The projection (M7) still has only finitely many deck terms by the
compact transporter of its cutoff support, regardless of whether
\(\Gamma\) itself is finitely generated. Its derivative commutator is
bounded by the same finite-support estimate. Every norm,
tensor-resolvent, block compression, graph-core and exact
derivative-cancellation calculation (M8)--(M14) consequently applies to
this self-adjoint direct sum. The quotient unitary maps onto the
sections over all base components exactly as before. This proves (M15)
with unchanged full/reduced and odd conventions. A finite-rank
coefficient twist changes only the already controlled component Diracs.
\(\square\)

\subsection{6. Models, lattices and trace
integrality}\label{models-lattices-and-trace-integrality}

\subsubsection{6.1. A proved universal comparison system and
independence of the
model}\label{a-proved-universal-comparison-system-and-independence-of-the-model}

Here \(G\) is an arbitrary second-countable locally compact group. A
proper \(G\)-space on which \(C_0\) and \(KK\) are used is locally
compact, Hausdorff and second countable; proper means that
\((g,x)\mapsto(gx,x)\) is proper. A universal ambient space need not
itself be locally compact. We will use only its closed, locally compact,
\(G\)-compact subspaces. This distinction is essential for the following
construction.

Let \[
 {\cal U}=\{u\in L^2(G,dg):u\text{ is real and nonnegative a.e.},\
                  \|u\|_2=1\},\qquad
 (\lambda_su)(g)=u(s^{-1}g).
 \tag{U1}
\] This is a closed subspace of the unit sphere of a separable Hilbert
space, with a continuous isometric action. Here are proofs of the
analytic assertions being used. Haar measure is finite on compact sets;
second countability gives a countable cover by relatively compact open
sets, hence a countable compact exhaustion and sigma-finiteness.
Regularity approximates a finite-measure measurable set from inside by a
compact set and from outside by an open set. A compactly supported
continuous function equal to one on the compact set and between zero and
one in that open set then approximates its indicator in \(L^2\).
Truncation, simple functions and the exhaustion prove density of
\(C_c(G)\) in \(L^2(G)\). Taking finite unions of a countable relatively
compact base, rational coefficients and this regular approximation
proves separability. For a function in \(C_c(G)\), continuity of
translation follows by uniform continuity on a compact neighborhood of
its support and integration there. Density and isometry give strong
continuity on \(L^2(G)\). These uses of compactly supported functions
and regular integration are exactly NCF.1, NCF.4 and NCF.5.

For \(u,v\in C_c(G)\), the coefficient \(\langle\lambda_gu,v\rangle\)
vanishes outside \((\operatorname{supp}v)(\operatorname{supp}u)^{-1}\).
Cauchy--Schwarz and \(L^2\)-approximation therefore give \[
 \langle\lambda_gu,v\rangle\longrightarrow0\quad(g\longrightarrow\infty)
 \tag{U2}
\] for arbitrary \(u,v\in L^2(G)\). The convergence is uniform when
\(u,v\) range over two fixed compact subsets: choose finite norm nets in
them and bound the difference of coefficients by the two approximation
errors times the bounded norms. In particular, if \(K,L\subset{\cal U}\)
are compact, the transporter \(\{g:gK\cap L\ne\varnothing\}\) is closed
and is contained in a compact set, since a matching pair has coefficient
one. Thus the action on \({\cal U}\) is proper: the inverse image of a
compact subset of \({\cal U}^2\) is closed in the product of a compact
transporter and a compact second-coordinate projection. This also proves
the needed local version: near a pair of unit vectors, continuity and a
fixed positive coefficient threshold bound all possible transporters.

For compact \(H\subset G\), choose a nonzero nonnegative
\(f\in C_c(G)\). Haar probability on \(H\) gives
\(v=\int_H\lambda_h f\,dh\). Its support is compact; it is \(H\)-fixed
and nonnegative, and \(\int_Gv=\int_Gf>0\), so it is nonzero. With
\(u_H=v/\|v\|_2\), the formula \[
 (u,t)\longmapsto
 \frac{(1-t)u+t u_H}{\|(1-t)u+t u_H\|_2}
 \tag{U3}
\] contracts \({\cal U}^H\). The denominator is at least \(1/\sqrt2\),
because inner products of nonnegative vectors are nonnegative. A
noncompact closed subgroup cannot fix a unit vector, since its
coefficient would be constantly one in (U2). In particular all
stabilizers are compact.

\textbf{Lemma A.7a (cutoff without cocompactness).} Every proper
\(G\)-space \(X\) in the stated category has a continuous
\(b:X\to[0,\infty)\) such that \[
 \int_G b(g^{-1}x)\,dg=1,\qquad
 \operatorname{supp}b\cap\pi^{-1}(L)\text{ is compact for every compact }
 L\subset X/G.
 \tag{U4}
\] Consequently \(F_b(x)(g)=\sqrt{b(g^{-1}x)}\) is a continuous
equivariant map \(X\to{\cal U}\).

\textbf{Proof.} First \(Y=X/G\) is Hausdorff, locally compact and second
countable. Properness makes the orbit relation closed: a convergent net
of related pairs is lifted into the inverse image of a compact
neighborhood of its limit, where a convergent subnet supplies the
limiting group element. The quotient map \(\pi\) is open. Its product
with itself is also an open quotient map, so the closed orbit relation
descends to a closed diagonal in \(Y^2\); this is the Hausdorff
assertion. Images of relatively compact open neighborhoods have compact
closures in \(Y\), and images of a countable base form a base. NCF.1
therefore supplies a locally finite partition of unity with compact
supports subordinate to any prescribed open cover, after refinement by
relatively compact open sets.

Choose nonnegative \(f_i\in C_c(X)\) whose positive sets \(W_i\) meet
every orbit, and put \[
 h_i(x)=\int_G f_i(g^{-1}x)\,dg,\qquad
 V_i=\pi(W_i).
\] The function \(h_i\) is invariant, finite and continuous: on a
compact neighborhood \(K\) of an argument, every contributing group
element lies in the compact transporter from \(\operatorname{supp}f_i\)
to \(K\). Integration thus takes place over one compact set. Positivity
on \(V_i\) follows from Haar positivity on a nonempty open set. Write
\(\bar h_i\) for the descended function on \(Y\), and choose a locally
finite partition \(\rho_i\) with compact support contained in \(V_i\),
allowing repeated indices after refinement. Define \[
 b(x)=\sum_i
       \frac{\rho_i(\pi x)}{\bar h_i(\pi x)}\,f_i(x).
 \tag{U5}
\] Each summand is extended by zero outside \(V_i\). This extension is
continuous because \(\operatorname{supp}\rho_i\) is compact inside
\(V_i\), where \(\bar h_i\) has a positive minimum. The sum is locally
finite over the quotient. Its integral is \(\sum_i\rho_i(\pi x)=1\), by
left invariance of Haar measure. Over a compact \(L\subset Y\), only
finitely many \(\rho_i\) occur; the support of \(b\) there is contained
in the finite union of their \(\operatorname{supp}f_i\). Being closed in
that compact union, it is compact.

The square root in (U4) has norm one. On a compact neighborhood
\(K\subset X\), only finitely many summands can contribute on the orbits
through \(K\); their contributing group elements again lie in one
compact transporter. Joint continuity of \(b(g^{-1}x)\) on that compact
set, followed by integration, proves norm continuity of \(F_b\). Finally
\(F_b(sx)(g)=F_b(x)(s^{-1}g)\), proving equivariance. \(\square\)

Any two continuous equivariant maps \(F_0,F_1:X\to{\cal U}\) have the
equivariant homotopy obtained by normalized convex interpolation, with
denominator at least \(1/\sqrt2\). This assertion applies to all such
maps, not only the cutoff maps.

\textbf{Lemma A.7b (the subspaces actually used by \(KK\)).} For every
compact \(K\subset{\cal U}\), its saturation \(Z_K=GK\) is closed,
locally compact, Hausdorff, second countable, proper and \(G\)-compact.

\textbf{Proof.} Uniform convergence in (U2) gives, for any
\(v\in{\cal U}\), a compact \(C\subset G\) such that
\(|\langle\lambda_gk,v\rangle|<1/2\) for \(g\notin C,\ k\in K\). If
\(\|\lambda_gk-v\|_2<1/2\), the real inner product is
\(1-\|\lambda_gk-v\|_2^2/2>7/8\), hence \(g\in C\). Thus \(GK\) near
\(v\) is contained in the compact image \(CK\). Every point in its
closure is consequently in \(CK\subset GK\); this proves closedness,
without an unproved assertion about arbitrary images of closed sets. For
\(v\in GK\), the closure in \(GK\) of its radius-\(1/4\) neighborhood is
contained in a compact image of this kind and is compact. This proves
local compactness. The remaining separation and countability properties
are inherited from the separable metric space \({\cal U}\); restriction
of a proper action stays proper. The quotient is the continuous image of
\(K\), so it is compact. \(\square\)

For later functoriality, an equivariant map from a \(G\)-compact proper
locally compact space \(X\) to any proper target \(T\) is proper,
provided the compact-transporter property holds in \(T\). Indeed choose
compact \(K\subset X\) with \(GK=X\), and let \(L\subset T\) be compact.
A point \(gk\) mapping into \(L\) has \(g\) in the compact transporter
from \(f(K)\) to \(L\). The closed preimage of \(L\) is therefore
contained in a compact image of this transporter times \(K\). This
proves the slight strengthening of A.3 that is needed here, including
\(T={\cal U}\). For a Hausdorff compactly generated target, this proper
map is closed: if \(C\subset X\) is closed and \(L\) is any compact
subset of the target, then \(f(C)\cap L=f(C\cap f^{-1}L)\) is compact
and thus closed in \(L\). The defining compact-test property gives
closedness in the target. Its image \(S\) is also locally compact and
second countable. Indeed the fiber over a point is compact; cover it by
finitely many relatively compact open subsets of \(X\). The complement
of the image of their closed complement is an open neighborhood in
\(S\), and its closure is contained in the compact image of their
finitely many compact closures. For second countability, take all finite
unions \(V\) of a countable base of \(X\); the sets
\(S\setminus f(X\setminus V)\) form a countable base, since the compact
fiber over a point of any open set can be covered by finitely many base
sets inside its inverse image. Its quotient is compact because it is the
image of \(f(K)\). This proves, rather than assumes, that all comparison
and homotopy images are legitimate locally compact stages. Metric
spaces, including \({\cal U}\), have the compact-test property. A
Hausdorff space with the weak topology of locally compact CW cell
domains also has it: a compact-test closed set has closed preimage in
each cell domain, since that domain is locally compact, hence compactly
generated; the final cell topology then makes the set closed. Thus this
formulation also permits the usual proper CW ambient models, without
asserting they are themselves locally compact.

\textbf{Theorem A.7 (model independence, including cycles and cutoffs).}
Define \[
 RK^G_i({\cal U};A)
   =\varinjlim_{K\subset{\cal U}\ {\rm compact}}
       KK^G_i(C_0(Z_K),A).
 \tag{U6}
\] The inclusions are proper and
\(Z_{K_1}\cup Z_{K_2}\subset Z_{K_1\cup K_2}\), so this is a directed
system. The maps in A.4 define an assembly map on (U6), independent of
all cutoff choices. For any Hausdorff compactly generated universal
proper ambient model \(E\), the direct limit over its closed
\(G\)-compact locally compact second-countable subspaces is canonically
isomorphic to (U6), and the assembly maps agree. All spaces used for
\(C_0\) and \(KK\) remain in the original proper-space category; the
preceding closed-image argument supplies those hypotheses for every
comparison image.

Here ``universal'' has its usual, explicit map-and-homotopy meaning:
every proper space in the category admits an equivariant map into \(E\),
and any two such maps are equivariantly homotopic. It suffices to
require this for \(G\)-compact spaces and their cylinders; no building
theorem or unproved existence result is invoked.

\textbf{Proof.} A cutoff map embeds each \(G\)-compact stage \(X\) into
the comparison system as a proper map with image in \(Z_{F_b(K)}\),
where \(GK=X\). A homotopy between any two maps has image in the
saturation of the compact set \(F(K\times[0,1])\). It is proper as a map
from \(X\times[0,1]\), by the same transporter argument. Therefore
source pullback gives the required pushforward
\(KK^G(C_0(X),A)\to KK^G(C_0(Z_{F_b(K)}),A)\); homotopy invariance says
that its class is independent of the map after passage to a common
stage. Restrictions to smaller stages and all choices are compatible by
the same homotopy. Conversely, for each \(Z_K\), universality gives a
map to \(E\); its image is a closed \(G\)-compact stage, and the same
argument makes the resulting direct-limit map independent and
compatible. Compositions are identities because any map from a stage to
its own model is homotopic, in a larger stage, to its inclusion. For
\({\cal U}\) this is the explicit convex homotopy; for \(E\) it is its
defining property. These statements use only proper maps and proper
homotopies between the locally compact stages, so their
\(C_0\)-pullbacks exist.

A.3--A.4 show equality of assembly under every one of those maps and
homotopies; A.2 proves independence of each cutoff projection. They
therefore give the canonical equality of assembly on the two limits. A
Kasparov homotopy or equality of cycle classes within any one stage is
already respected by descent and Kasparov product, and equality arising
only after a larger stage is respected by the just-proved compatibility.
This proves cycle independence as well as model independence, for both
full and reduced assembly and with coefficients. \(\square\)

For \(\Gamma=\mathbb Z^n\), its translation action on \(\mathbb R^n\) is
a universal model in precisely this category. To prove the defining
property, take (U4) for any proper \(\Gamma\)-space and set \[
 f(x)=\sum_{m\in\mathbb Z^n}b(m^{-1}x)m\in\mathbb R^n.
 \tag{U7}
\] The sum is locally finite: on a compact neighborhood of \(x\), the
proper-over-quotient support in (U4) and the compact-transporter
property allow only finitely many \(m\). Its weights sum to one, so
\(f(sx)=s+f(x)\). Any two equivariant maps to this affine space have the
straight-line equivariant homotopy. Translation is proper, and the
standard compact cube has saturation \(\mathbb R^n\). This proves the
asserted model and shows that its whole space is already a cofinal
stage: \[
 RK^\Gamma_i(E\Gamma;A)=KK^\Gamma_i(C_0(\mathbb R^n),A).
 \tag{U8}
\] A torsion-free discrete group has no nontrivial compact stabilizers
in a proper space: its stabilizers are finite by properness and
discreteness, and finite subgroups of a torsion-free group are trivial.
Thus for such a group the construction uses free proper spaces, as
required for the covering formulation below.

\textbf{Source and scope.} Wolfgang Lück, \emph{Survey on classifying
spaces for families of subgroups}, arXiv:math/0312378v2, Definitions 1.8
and 2.1 and Theorem 2.5, gives the free-source formulation of the
proper/numerable distinction and universal map-and-homotopy formulation.
The entire comparison proof actually used here is (U1)--(U8), including
the local compactness of the stages; no cited theorem replaces it.

\subsubsection{6.2. Free quotients and assembly on all analytic
cycles}\label{free-quotients-and-assembly-on-all-analytic-cycles}

This section treats a countable discrete group \(\Gamma\), a free proper
\(\Gamma\)-compact space \(X\), and \(Y=X/\Gamma\). Put
\(D=C_0(X)\rtimes\Gamma\). The assertions are not restricted to Dirac
classes.

\textbf{Lemma A.8a (dual Green--Julg, with its proof).} For a separable
\(\Gamma\)-algebra \(B\), with trivial action on the scalar coefficient,
\[
 KK_i^\Gamma(B,\mathbb C)\cong KK_i(B\rtimes\Gamma,\mathbb C).
 \tag{Q1}
\] A cycle \((H,\phi,F,U)\) is sent to \((H,\phi\rtimes U,F)\).

\textbf{Proof.} For a nondegenerate coefficient representation,
integration and the full crossed-product universal property give
mutually inverse correspondences between representations of
\(B\rtimes\Gamma\) and covariant pairs \((\phi,U)\). For a generator
\(aU_g\), the commutator is
\([F,\phi(a)]_{\rm gr}U_g+(-1)^{|a|}\phi(a)[F,U_g]\), for homogeneous
\(a\). Both terms are compact by the equivariant cycle conditions. The
other defects are compact after multiplication by this generator for the
same reason. Finite sums are dense; their operator-norm limits preserve
the compact conditions. Conversely, the crossed-product conditions on
\(aU_e\) give the ordinary cycle conditions, and the commutator on
\(aU_g\), minus the already compact first term, gives the localized
equivariance defect. Strong continuity has no additional requirement for
a discrete group. Essential replacement, proved in the product lesson,
permits nonessential initial representations; the strict multiplier
representation supplies the group unitaries on the essential subspace.
These operations agree under orthogonal sums, degenerate additions,
locally compact changes and cycle homotopies. The same calculation on
Hilbert \(C([0,1])\)-modules proves the homotopy assertion. Ordered
right Clifford factors commute with all group unitaries, giving both
degrees. This proves (Q1), not merely a correspondence between a
selection of geometric cycles. \(\square\)

\textbf{Lemma A.8b (the free-action Morita equivalence).} Full and
reduced \(D\) agree. If \(c\) is the normalized cutoff in A.1, its
projection \(p\) is full, and \[
 pDp=C(Y)p\cong C(Y).
 \tag{Q2}
\] An explicit \(D\)-\(C(Y)\) imprimitivity module \(E\) is the
completion of \(C_c(X)\), with \[
 \begin{split}
 (\xi h)(x)&=\xi(x)h([x]),\\
 \langle\xi,\eta\rangle([x])
   &=\sum_{\gamma}\overline{\xi(\gamma^{-1}x)}\eta(\gamma^{-1}x),\\
 (f\xi)(x)&=\sum_{\gamma}f(\gamma,x)\xi(\gamma^{-1}x).
 \end{split}
 \tag{Q3}
\]

\textbf{Proof.} A free proper discrete action has evenly covered local
charts. At \(x\), first take a relatively compact neighborhood.
Properness bounds its possible self-transporters by a finite set.
Freeness separates \(x\) from its images under the finitely many
nonidentity elements; shrinking the neighborhood separates all of those
images, while the other elements were already excluded. Its translates
are pairwise disjoint. This proves the chart assertion locally and
proves that \(X\to Y\) is a covering.

The kernel \[
 k_{\xi,\eta}(\gamma,x)
       =\xi(x)\overline{\eta(\gamma^{-1}x)}
 \tag{Q4}
\] has compact support and finite group support by properness. Direct
summation gives \(k_{\xi,\eta}^*=k_{\eta,\xi}\) and
\(k_{\xi,\eta}k_{\zeta,\omega}
 =k_{\xi\langle\eta,\zeta\rangle,\omega}\). Moreover these kernels span
the crossed-product core exactly. Indeed a core function has only
finitely many \(\gamma\)'s and a compact \(x\)-support. Partition that
support among finitely many of the free charts \(V_j\). For its term at
\(\gamma\), choose \(\xi_j\) to be the partitioned coefficient and
choose \(\eta_j\in C_c(\gamma^{-1}V_j)\) equal to one on
\(\gamma^{-1}\operatorname{supp}\xi_j\). The kernel \(k_{\xi_j,\eta_j}\)
is zero at every group element other than \(\gamma\): otherwise a
nonidentity translate of \(V_j\) would meet \(V_j\). At \(\gamma\) it is
exactly the partitioned coefficient. Summing proves the spanning
assertion.

The orbit sum in (Q3) is positive and continuous, and locally finite in
a covering chart; it vanishes only when the vector is zero. Thus it
completes \(C_c(X)\) to a Hilbert \(C(Y)\)-module. Multiplication by
\(a\in C_0(X)\) is bounded by \(\|a\|\), and translations are unitary;
this proves that the covariant representation integrates in the full
norm. Its kernels (Q4) act as the rank-one operators of (Q3).

Since \(\langle c,c\rangle=1\), \(p=k_{c,c}\), and the kernel identities
give \(k_{\xi,c}\,p\,k_{c,\eta}=k_{\xi,\eta}\). The core-spanning
assertion proves fullness of \(p\) in the full algebra. Also
\(p f p=h_fp\), where \(h_f=\langle c,fc\rangle\in C(Y)\). Conversely
every \(hp\), \(h\in C(Y)\), is in \(pDp\): invariant \(h\) is a bounded
multiplier and \(p\) has compact support. In every covariant
representation \(\|hp\|\leq\|h\|\); the orbit representation on
\(\ell^2(\Gamma)\) gives the reverse inequality at each orbit, since
\(p\) is its rank-one projection onto the unit vector \(c\). Thus the
full norm of \(hp\) is exactly \(\|h\|\). Completion gives (Q2).

The map \(\xi\mapsto k_{\xi,c}\) identifies (Q3) isometrically with
\(Dp\), since its inner product is \(\langle\xi,\eta\rangle p\). The
spanning assertion and the kernel identities give dense range. The
full-corner construction now proves imprimitivity directly: rank-one
operators on \(Dp\) are left multiplication by \(DpD\), dense in \(D\),
and a multiplier annihilating \(Dp\) annihilates that dense ideal and is
zero. The dual module \(pD\), with the evaluation maps
\(d_1p\otimes pd_2\mapsto d_1pd_2\), gives both inverse KK classes by
the actual interior-tensor identities.

Finally the regular quotient is faithful on \(pDp=C(Y)\), by the same
orbit representations. If its kernel is \(I\), then \(pIp=0\). For
\(a\in I\), \(pa^*ap=0\) gives \(ap=0\); applying this to \(ad\in I\)
for every \(d\in D\) gives \(aDp=0\). Fullness implies \(aD=0\), hence
\(a=0\). This proves full equals reduced here, without a general
amenability theorem. \(\square\)

Combining (Q1)--(Q3) gives an actual isomorphism \[
 \Theta_X:KK_i^\Gamma(C_0(X),\mathbb C)
              \longrightarrow KK_i(C(Y),\mathbb C),\qquad
 \Theta_X(x)=[E^*]\otimes_D(\phi\rtimes U,F).
 \tag{Q5}
\] For an invariant covering Dirac, compression by \(p\), with scalar
coefficients, is the quotient Dirac: (M9)--(M14), with \(A=\mathbb C\),
give its full source action and closure. Consequently
\(\Theta_X[D_X]=[D_Y]\), with precisely the odd and grading conventions
already fixed there.

Let \(A=C^*(\Gamma)\), or \(C_r^*(\Gamma)\), and let
\(\mathcal L=X\times_\Gamma A\) use the left-unitary action \(u_\gamma\)
and the usual right \(A\)-module structure. The locally constant
transitions and a finite chart partition of the compact \(Y\) make its
section module finitely generated projective over \(C(Y)\otimes A\): the
local components weighted by square roots of a partition embed it
isometrically in a finite free module, and the adjoint patches them
back. Denote its scalar KK class by \([\mathcal L]\).

In this subsection write \(\mu_X^A\) for \textbf{scalar-coefficient}
assembly with target completion \(A=C^*(\Gamma)\) or \(C_r^*(\Gamma)\),
and \(j_\Gamma^A\) for that scalar cycle's full or reduced descent. Thus
\(\mu_X^{C^*(\Gamma)}=\mu_{X,\mathbb C}\) and
\(\mu_X^{C_r^*(\Gamma)}=\mu_{X,\mathbb C,r}\) in the notation of A.4.
The superscript selects the target completion; it does not change the
input coefficient to \(A\).

\textbf{Theorem A.8 (Mishchenko comparison for every analytic cycle).}
For every \(x\) in (Q5), writing \(y=\Theta_X(x)\), \[
 \mu_X^A(x)
    =[\mathcal L]\otimes_{C(Y)\otimes A}(y\boxtimes1_A).
 \tag{Q6}
\] This includes both degrees, arbitrary analytic cycles, and
homotopies.

\textbf{Proof.} The covariant pair
\((a\mapsto a\otimes1,\ U_\gamma\mapsto
U_\gamma\otimes u_\gamma)\) defines a homomorphism \[
 \delta:D\longrightarrow D\otimes A,\qquad
 \delta(aU_\gamma)=aU_\gamma\otimes u_\gamma .
 \tag{Q7}
\] Existence in the minimal tensor product follows directly from the
full universal property into its multiplier algebra; core terms are in
\(D\otimes A\), and their dense closure gives the displayed range.

On the crossed Hilbert module of a scalar cycle, the map from compactly
supported vectors to \(H\otimes A\) is
\(\zeta\mapsto\sum_\gamma\zeta(\gamma)\otimes u_\gamma\). The descent
inner product becomes
\(\sum_{\gamma,\eta}\langle\zeta(\gamma),\zeta'(\eta)\rangle
u_\gamma^*u_\eta\), exactly the tensor inner product. Its range is
dense; it intertwines right multiplication, the descended operator
\(F\otimes1\), and the left generator with
\(\phi(a)U_\gamma\otimes u_\gamma\). Thus \[
 j_\Gamma^A(x)=[\delta]\otimes_{D\otimes A}
                         ((\phi\rtimes U,F)\boxtimes1_A).
 \tag{Q8}
\] This is also a direct cycle proof of the comparison, rather than an
imported coaction formula.

Transport the projection \(\delta(p)\) through \(E\boxtimes A\). On the
module of functions in (Q3) with \(A\)-coefficients, it is \[
 (PZ)(x)=c(x)\sum_\gamma c(\gamma^{-1}x)
                   u_\gamma Z(\gamma^{-1}x).
 \tag{Q9}
\] A section of \(\mathcal L\) lifts to
\(\Psi(\gamma x)=u_\gamma\Psi(x)\). The map \(\Psi\mapsto c\Psi\) lands
in the range of \(P\), and its pointwise orbit inner product is
\(\sum_\gamma c(\gamma^{-1}x)^2\Psi(x)^*\Phi(x)=\Psi(x)^*\Phi(x)\). Its
inverse on the range is the sum in (Q9) without the outer \(c(x)\).
These sums are locally finite, and the range vectors have support in the
compact support of \(c\); they give inverse adjointable unitaries,
commuting with \(C(Y)\). Hence \[
 [\delta(p)]\otimes_{D\otimes A}[E\boxtimes A]=[\mathcal L].
 \tag{Q10}
\] The Morita inverse identities give
\((\phi\rtimes U,F)=[E]\otimes_{C(Y)}y\). Substitute this into (Q8),
multiply by \([p]\), and use (Q10) and associativity. The result is
(Q6). All operators and maps commute with the last right Clifford
factor, so the statement holds in both degrees. They also act on
interval-coefficient modules, proving the assertion about every analytic
homotopy. \(\square\)

When \(Y\) is a closed spin\(^{c}\) manifold and \(a\in K_j(C(Y))\), let
\(a\cap[D_Y]\in KK_{j+\dim Y}(C(Y),\mathbb C)\) mean the usual action
through diagonal multiplication. Formula (Q6), including the
\(C(Y)\)-linear unitary in (Q10), proves \[
 \mu_X^A\bigl(\Theta_X^{-1}(a\cap[D_Y])\bigr)
                =a\otimes_{C(Y)}[D_{\mathcal L}].
 \tag{Q11}
\] For finite-rank bundle representatives this is the tensor connection
Dirac and the full-source version of (M13). For odd representatives it
follows from the same diagonal action, (Q6), and associativity; it is
not an assertion that all odd representatives are bundles. Product
recognition for the Dirac action is the actual Theorem 5.4 of Lesson 11;
all the module and closure assertions are those of M.3. Thus (Q11)
retains the scalar and odd source actions needed for the torus
calculation.

\textbf{Source.} The freely available primary author paper by Markus
Land, arXiv:1306.5657v2, Section 4, Propositions 4.1--4.5 give the
free-source comparison. Its quoted external proof of (Q1) is not used as
a provider: that result is proved in A.8a. Likewise (Q2)--(Q11) supply
the full local corner, module and coaction arguments actually used here.

\subsubsection{6.3. Lattice assembly and its ordered Connes--Thom
comparison}\label{lattice-assembly-and-its-ordered-connesthom-comparison}

Write \(T^n=(\mathbb R/\mathbb Z)^n\), order the positive coordinates
\(x_1,\ldots,x_n\), and use periodic spinors. Let \(D_n\) be the
ordinary positive-volume spin\(^{c}\) Dirac of the flat torus. For
\(n=1\), \(D_1=-i\,d/dx\). The full-source covering cycle of M.3 is
\(d_n=[D_{\mathcal L_n}]\in KK^n(C(T^n),C^*(\mathbb Z^n))\). All integer
degrees below use the ordered right-Clifford convention of the preceding
lessons.

\textbf{Lemma A.9a (the torus duality actually needed here).} The map \[
 \operatorname{PD}_n:K_j(C(T^n))\longrightarrow
 KK_{j+n}(C(T^n),\mathbb C),\qquad
 a\longmapsto a\cap[D_n],
 \tag{T1}
\] is an isomorphism for every \(n\geq0\).

\textbf{Proof.} This is proved for the torus, without assuming the
general manifold duality statement. Evaluation at \(1\) on the circle
has the homomorphic section of constants and kernel
\(C_0(S^1\setminus\{1\})\cong C_0(\mathbb R)\). The actual semisplit
excision and both-variable exact sequences,
\href{KT-KK-14.html\#4-excision-and-the-six-term-sequences}{Lesson 14,
Theorems 4.1--4.2}, apply to its tensor product with every \(C(T^r)\):
pointwise evaluation and the same constants section give the split
extension explicitly. Its extension cycle is zero, since a homomorphic
section makes its Stinespring projection commute and its commutator
operator zero; equivalently its direct split extension has a degenerate
boundary cycle. Thus both K-theory and contravariant scalar KK split
into the base summand and its suspended summand. The fixed Bott inverse
cycles give the suspension identifications.
\href{KT-KK-07.html\#2-the-even-scalar-fredholm-picture}{Lesson 07,
Theorems 2.2 and 3.2}, gives \(KK_0(\mathbb C,\mathbb C)=\mathbb Z\) by
rank/index and \(KK_1(\mathbb C,\mathbb C)=0\); finite-dimensional
unitary diagonalization also proves the latter scalar K-group is zero.

For the circle let \(z(x)=e^{2\pi ix}\), \(e\) be evaluation at \(1\),
and \(d=[D_1]\). The reduced class of \(z\) is a generator of
\(K_1(C_0(S^1\setminus\{1\}))\): in the increasing coordinate
\(\xi=-\cot(\pi x)\) it is exactly the positive Cayley generator
\((\xi-i)/(\xi+i)\) of the scalar Bott proof in Lesson 18, Section 5.
The positive circle Dirac compresses \(z\) to the unilateral shift. Its
kernel is zero and its cokernel has dimension one, so
\(\langle[z],d\rangle=-1\), the actual calculation (CI.39) of Lesson 15.
This pairing with the reduced generator proves that \(d\) generates the
odd suspended KK summand: composing restriction with the fixed Bott
inverse identifies that summand with \(\mathbb Z\), and this pairing is
exactly its integer coordinate. The even group is generated by \(e\).
Consequently \[
 1\cap d=d,\qquad [z]\cap d=-e.
 \tag{T2}
\] For the second identity both sides lie in the even group
\(\mathbb Z e\), and pairing with the scalar unit computes their
coefficient as \(-1\).

Iterate the split extension. Naturality of restriction and the fixed
suspension isomorphisms shows inductively that exterior products of
\(1,[z]\) form an integral basis of \(K_*(C(T^n))\), and exterior
products of \(e,d\) form an integral basis of its scalar KK groups.
Indeed restriction to the new suspended ideal carries
\([z]\boxtimes b\), or \(d\boxtimes b\), to the suspension of \(b\), up
to the fixed invertible Bott sign; the constant/evaluation factors give
the other split summand. This proves surjectivity and independence of
these products at each step, not a Künneth assumption.

Set \(\epsilon_n=(-1)^{n(n-1)/2}\). The operator and grading calculation
of \href{KT-KK-13.html\#7-pullback-and-direct-sums}{Lesson 13,
Proposition 7.2}, applies to the ordered derivatives as follows. Their
squares add on the Fourier core, so the sum is self-adjoint on the
domain weighted by \(1+\sum k_j^2\), has compact resolvent, and its
domain is contained in each derivative domain. Smooth finite Fourier
creations have bounded derivative errors; the anticommutation gives the
adjoint errors and first-factor positivity \(2D_{\rm first}^2\geq0\).
The unbounded product criterion of
\href{KT-KK-11.html\#5-recognizing-unbounded-products}{Lesson 11,
Theorem 5.4}, therefore identifies that sum with the exterior product.
In two odd factors the product matrices are \(\sigma_1,-\sigma_2\),
whereas the ordinary direct sum uses \(\sigma_1,\sigma_2\); conjugation
by \(\sigma_1\) reverses the grading and changes between them. In mixed
parity the ordinary graded sum agrees. Iteration gives, with the integer
rank retained, \[
 [D_n]=\epsilon_n\,d\boxtimes_R\cdots\boxtimes_R d.
 \tag{T3}
\] The same calculation works for the flat \(A\)-module twists, with
compactness and domains supplied by M.3.

For a subset \(S\subset\{1,\ldots,n\}\), let \(a_S\) be the exterior
product with \(z\) in \(S\) and \(1\) elsewhere, and let \(b_T\) have
\(d\) in \(T\) and \(e\) elsewhere. Move each odd \(d\) past the later
odd \(z\)'s using the proved graded exterior interchange law, then use
(T2). The result is \[
 \operatorname{PD}_n(a_S)
       =\epsilon_n(-1)^{\sum_{s\in S}s}\,b_{S^c}.
 \tag{T4}
\] This is a signed permutation of integral bases, proving (T1) in both
degrees. For \(n=0\) it is the identity. \(\square\)

We next identify \(d_n\) with the invertible Thom correspondence itself.
The sign and the target coordinate convention are part of this
identification. On \(B=C(\mathbb R/\mathbb Z)\) use translation
\(\alpha_t f(x)=f(x-t)\). The actual Green theorem,
\href{../KT-CP/KT-CP-07.html\#completion-in-the-full-crossed-product-norms}{KT-CP
Lesson 07, Lemmas 7.1--7.2, Proposition 7.3 and Theorem 7.4}, proves
positivity, full norm completion and imprimitivity; its transported
formula (7.38), for \(G=\mathbb R,H=\mathbb Z,A=\mathbb C\), gives the
module \(E_+\) with \[
 (\xi u_m)(r)=\xi(r-m),\qquad
 \langle\xi,\eta\rangle_+(m)=
       \int_{\mathbb R}\overline{\xi(r)}\eta(r+m)\,dr.
 \tag{T5}
\] Left functions multiply and left group elements translate by
\(r\mapsto r-t\). Its source is \(B\rtimes_\alpha\mathbb R\).

For the Mishchenko convention (M8), use the equally invertible
\textbf{holonomy-compatible} Green module \(E_-\), obtained by the
target automorphism \(\iota(u_m)=u_{-m}\). Explicitly \[
 (\xi\cdot u_m)(r)=\xi(r+m),\qquad
 \langle\xi,\eta\rangle_-(m)
       =\int_{\mathbb R}\overline{\xi(r)}\eta(r-m)\,dr.
 \tag{T6}
\] The right action is the old action by \(\iota(u_m)\) and the inner
product is \(\iota^{-1}\) of the old one; hence positivity, completion
and imprimitivity follow by an isometric change of coefficient algebra,
with inverse \(\iota\). This proves the change rather than ignoring a
holonomy inversion.

The map \[
 (\mathcal P\xi)(x)=\sum_{m\in\mathbb Z}\xi(x+m)u_{-m}
 \tag{T7}
\] extends to a unitary from \(E_-\) onto the \(L^2\) section module of
\(\mathcal L_1\). The sum is locally finite on its initial compactly
supported domain, and \((\mathcal P\xi)(x+1)=u_1(\mathcal P\xi)(x)\). It
intertwines the right action in (T6), as follows by reindexing \(m\).
Expanding the inner product and integrating over \(0\leq x\leq1\) gives
the coefficient at \(u_k\) equal to
\(\int_{\mathbb R}\overline{\xi(r)}\eta(r-k)\,dr\), exactly (T6). Finite
local trivializations and finite group-algebra coefficients approximate
every section in norm; their scalar compactly supported lifts times
right coefficients belong to the range, proving density and hence
surjectivity. The unitary retains the left action of \(f(x)\), and left
translation becomes \(\Psi(x)\mapsto\Psi(x-t)\).

In Lesson 18 the canonical group multiplier is \(u_t=e^{itP_\alpha}\).
Since \(D_{\mathcal L_1}=-i\,d/dx\), the last translation is
\(e^{-itD_{\mathcal L_1}}\); therefore \(P_\alpha=-D_{\mathcal L_1}\)
under (T7). Choose the odd bounded transform \(h(s)=s/\sqrt{1+s^2}\) for
the Thom cycle. The full-source cycle identification is \[
 t_\alpha\otimes_{B\rtimes\mathbb R}[E_-]
       =-[D_{\mathcal L_1}]\quad\hbox{in }KK^1(B,C^*(\mathbb Z)).
 \tag{T8}
\] The closures and graph cores agree under (T7), first on smooth
compactly supported lifts and then by the self-adjoint regular closures
proved in M.3. Negative \(h(D)\) represents the additive inverse in the
exact odd picture of Lesson 07. Thus (T8) includes the operator, source
action and odd sign.

For \(n\) coordinates, independent applications of the same proved Green
theorem give the exterior module \(E_-^{(n)}\). The tensor maps on
compactly supported product functions preserve the iterated inner
products by Fubini and have dense range by compact-support partitions.
They identify the iterated crossed product with
\(C(T^n)\rtimes\mathbb R^n\), and \(E_-^{(n)}\) is its imprimitivity
module to \(C^*(\mathbb Z^n)\), with the target automorphism
\(u_m\mapsto u_{-m}\) in every coordinate. These iterated covariant
integrated maps are inverse on generators, so no unproved
tensor/crossed-product identification is required. The multidimensional
version of (T7) is the same periodization sum over \(m\in\mathbb Z^n\).

Let \(t_{\downarrow}^{(n)}\) denote the iterated KK Thom element
crossing in the order \(e_n,e_{n-1},\ldots,e_1\), the precise ordered
convention of
\href{../KT-CP/KT-CP-11.html\#iteration-in-ordered-coordinates}{KT-CP
Lesson 11}. Inactive-factor naturality, proved in
\href{KT-KK-18.html\#4-the-real-thom-operator}{Lesson 18, Proposition
4.2}, and the graded exterior flip show that
\(t_{\downarrow}^{(n)}=\epsilon_n(t_{\alpha_1}\boxtimes_R\cdots\boxtimes_Rt_{\alpha_n})\)
under the just-described product-coordinate identification. Reversing
\(n\) odd factors makes \(n(n-1)/2\) odd transpositions. Equations (T3)
and (T8), including their identical matrix calculation for the flat
twists, now give \[
 d_n=(-1)^n\,t_{\downarrow}^{(n)}
           \otimes_{C(T^n)\rtimes\mathbb R^n}[E_-^{(n)}].
 \tag{T9}
\] Both \(\epsilon_n\)'s cancel. The target inversion built into
\(E_-^{(n)}\) is indispensable; (T9) with the usual \(E_+\) and
unchanged target coordinates would be a different formula.

\textbf{Theorem A.9 (lattice assembly).} For every \(n\geq0\), full and
reduced assembly for \(\mathbb Z^n\) are isomorphisms in both degrees.
More precisely, for \(a\in K_j(C(T^n))\), \[
 \mu_{j+n}\!\left(\Theta_{\mathbb R^n}^{-1}
                       (\operatorname{PD}_n(a))\right)
     =(-1)^n\,{\mathcal G}_{-,*}\,
                         \Gamma_{\alpha,\downarrow}^{(n),j}(a).
 \tag{T10}
\] Here \(\Gamma\) is the actual KK Connes--Thom map of Lesson 18 and
\({\mathcal G}_{-,*}\) is the isomorphism induced by the specified
holonomy-compatible Green module.

\textbf{Proof.} The proper universal model and its cofinal whole-space
stage are (U7)--(U8). Formula (Q5) identifies its entire source with
scalar K-homology of \(T^n\). Lemma A.9a identifies that entire group
with \(K_{*-n}(C(T^n))\). For every one of those classes (Q11), followed
by the full-source correspondence equality (T9), proves (T10). The KK
Thom element is invertible by the actual Fack--Skandalis proof of
\href{KT-KK-18.html\#6-the-fack-skandalis-argument}{Lesson 18, Theorem
6.1}; its ordered iteration is invertible. Green imprimitivity and
multiplication by \((-1)^n\) are also invertible. Therefore the assembly
map itself is an isomorphism.

For completeness, the full and reduced group algebras of \(\mathbb Z^n\)
agree with \(C(T^n)\). The full universal algebra of \(n\) commuting
unitaries is \(C(T^n)\): Laurent polynomials are uniformly dense, by
successive one-variable Fourier convolution (or the compact commutative
approximation theorem), and evaluation at each character supplies the
exact supremum norm. Fourier series identify \(\ell^2(\mathbb Z^n)\)
with \(L^2(T^n)\); the regular unitaries become multiplication by the
coordinate functions. This representation is faithful because Haar
measure has full support and a nonzero continuous function has nonzero
norm as a multiplication operator. Thus the regular quotient is
injective. Full-to-reduced compatibility in A.4 finishes the reduced
assertion with the same coordinates. \(\square\)

To translate (T10) into the positive-in-both-parities convention
\(\Phi\) of
\href{../KT-CP/KT-CP-11.html\#iteration-in-ordered-coordinates}{KT-CP
Lesson 11}, recall the actual comparison proved in
\href{KT-KK-18.html\#6-the-fack-skandalis-argument}{Lesson 18,
Proposition 6.2}: \(\Gamma_\alpha^j=(-1)^j\Phi_\alpha^j\). During \(n\)
steps the input parities are \(j,j+1,\ldots,j+n-1\), so \[
 \Gamma_{\alpha,\downarrow}^{(n),j}
     =(-1)^{nj+n(n-1)/2}\Phi_{\alpha,\downarrow}^{(n),j},
 \qquad
 \mu(\Theta^{-1}\operatorname{PD}_n(a))
     =(-1)^{n+nj+n(n-1)/2}
                  {\mathcal G}_{-,*}\Phi_{\alpha,\downarrow}^{(n),j}(a).
 \tag{T11}
\] This retains integer rank, rather than dropping \(\epsilon_n\) after
reduction modulo two.

\textbf{The assigned two-Thom exercise.} For \(n=2\), first cross the
\(x_2\)-translation and then the \(x_1\)-translation. Each is an
isomorphism, their product has degree two, and (T11) says in both
parities \[
 \mu_i\bigl(\Theta^{-1}\operatorname{PD}_2(a)\bigr)
       =-{\mathcal G}_{-,*}
                 \Phi_{\alpha_1}^{\,j+1}\Phi_{\alpha_2}^{\,j}(a),
 \qquad i=j\pmod2.
 \tag{T12}
\] The quotient identification and the signed-permutation duality (T4)
are isomorphisms; hence (T12) proves the required \(\mathbb Z^2\)
assembly isomorphism by two actual Connes--Thom maps. Its cutoff
projection is the product tent cutoff of A.6. The physical torus Dirac
has matrix \(-i(\sigma_1\partial_{x_1}+\sigma_2\partial_{x_2})\) and
grading \(\sigma_3\), whereas the ordered two-circle product has the
minus before \(\sigma_2\). Equations (T3), (T8) and the holonomy
inversion (T6) explain every sign in (T12).

\subsubsection{6.4. Integral trace on the entire torsion-free assembly
image}\label{integral-trace-on-the-entire-torsion-free-assembly-image}

This proof covers all analytic cycles, without assuming their generation
by manifolds or a general Poincaré duality theorem. Its inputs are
A.7--A.8, the actual scalar pictures, Bott inverse and semisplit
excision, and the finite-complex arguments proved next. A tracial Chern
character is used only on finite complexes, not on arbitrary compact
Hausdorff spaces.

\textbf{Lemma A.10a (finite-complex reduction).} Every free proper
\(\Gamma\)-compact space \(X\), for countable discrete \(\Gamma\),
admits an equivariant map to a free proper \(\Gamma\)-compact locally
finite simplicial space \(Z\) whose quotient is a finite polyhedron. The
map of quotient spaces pulls the flat Mishchenko bundle on \(Z/\Gamma\)
back to that on \(X/\Gamma\).

\textbf{Proof.} Use the free charts proved in A.8b. Compactness of
\(X/\Gamma\) permits finitely many nonnegative \(f_i\in C_c(X)\),
supported in such charts, whose positive translates cover \(X\). Put \[
 h(x)=\sum_{i,\gamma}f_i(\gamma^{-1}x)^2,\qquad
 t_{\gamma,i}(x)=f_i(\gamma^{-1}x)^2/h(x).
 \tag{R1}
\] The denominator is positive, invariant and continuous; properness
makes the sums locally finite. For each \(i\), at most one
\(t_{\gamma,i}\) is nonzero, since the translated chart supports are
disjoint. These barycentric coordinates define a continuous equivariant
map into the join of finitely many copies of the discrete set
\(\Gamma\), with vertex tags \(i\). Continuity holds in the simplicial
topology: near each \(x\) only finitely many \(\gamma\)'s can occur, by
the compact-transporter argument, so the map locally takes values in a
finite subcomplex.

Only finitely many orbits of simplices occur. If both \((\gamma,i)\) and
\((\eta,j)\) occur, then \(\gamma^{-1}\eta\operatorname{supp}f_j\) meets
\(\operatorname{supp}f_i\); there are finitely many such elements. After
translating the first occurring vertex to \((e,i)\), all the other
labels belong to these fixed finite lists. Take all those simplices and
their faces, and all their translates, to form \(Z\). A stabilizer of a
simplex fixes its uniquely tagged vertices, so it is trivial. Every
vertex belongs to finitely many simplices: there are finitely many orbit
representatives, and translating a specified vertex of one
representative to the given vertex determines the translating element
uniquely. Thus \(Z\) is locally finite and locally compact. A compact
subset meets finitely many closed simplices, by a finite cover of its
finite-star neighborhoods. Transporters of two such compact subsets are
finite, because matching two finite sets of vertices determines only
finitely many translations. Hence the action is proper.

The quotient has finitely many cells, and no cell closure identifies two
of its own faces: the vertex tags in a simplex are all distinct, and any
equality of a face with its translate fixes one vertex and hence has
translating element \(e\). Barycentric subdivision, with vertices the
actual cells and simplices their inclusion chains, consequently
triangulates this finite quotient (distinct cells give distinct new
vertices). It is a finite polyhedron \(P=Z/\Gamma\). The same
subdivision upstairs retains a proper covering \(Z\to P\). The
constructed map \(f:X\to Z\) is proper by A.3; on each orbit it is a
bijection onto the orbit of its image. Therefore the map
\((x,a)\mapsto(f(x),a)\) gives the asserted isometric identification of
the pulled-back principal bundle and its associated \(A\)-module bundle.
This also follows directly in the free local charts. \(\square\)

Here are the finite-complex algebra and character facts used in the
trace argument.

\textbf{Lemma A.10b (rational external-product decomposition, proved by
cells).} For a finite CW complex \(P\) and any unital C*-algebra \(A\),
external product gives an isomorphism \[
 \left(K_0(C(P))\otimes K_0(A)\ \oplus\
       K_1(C(P))\otimes K_1(A)\right)\otimes\mathbb Q
          \ \xrightarrow{\ \cong\ }\
                    K_0(C(P)\otimes A)\otimes\mathbb Q ,
 \tag{R2}
\] and the corresponding odd decomposition. Scalar K-groups of \(P\) are
finitely generated.

\textbf{Proof.} For a closed subcomplex \(F\subset P\), restriction
\(C(P)\to C(F)\) has a completely positive contractive linear section.
One explicit construction, for a compatible compact metric, is this. On
\(P\setminus F\), choose a locally finite partition on balls with
centers \(x_i\) and radius smaller than
\(\operatorname{dist}(x_i,F)/3\). Choose \(a_i\in F\) with distance less
than \(2\operatorname{dist}(x_i,F)\). Set
\(\sigma f(x)=\sum_i\rho_i(x)f(a_i)\) off \(F\), and \(f(x)\) on \(F\).
For a point in one of those balls,
\(\operatorname{dist}(x,a_i)\leq(7/2)\operatorname{dist}(x,F)\); uniform
continuity of \(f\) proves continuity at \(F\). Convex sums prove the
norm bound, unitality and positivity of every matrix amplification,
hence complete positivity. The case \(F=\varnothing\) is immediate.
NCF.1 supplies the required partition.

Tensoring this extension with \(A\) preserves its kernel. Indeed
\(\sigma\otimes1_A\) is completely positive contractive, and
\((1-\sigma q)\otimes1_A\) is bounded and takes finite sums into
\(C_0(P\setminus F)\otimes A\). On the kernel of \(q\otimes1_A\) it is
the identity; approximating by finite sums proves that this kernel is
exactly the stated ideal. The tensor extension is therefore semisplit.
Its boundary is the exterior tensor of the scalar boundary with \(1_A\):
tensor the actual Stinespring module and commutator cycle of
\href{KT-KK-14.html\#3-semisplit-extensions-and-their-classes}{Lesson
14, Section 3}, to see this directly. Thus external products commute
with the two exact sequences of
\href{KT-KK-14.html\#4-excision-and-the-six-term-sequences}{Lesson 14,
Theorem 4.2}, with precisely its graded interchange signs.

Apply those sequences to successive skeleta of \(P\). Each relative
ideal is a finite direct sum of \(C_0(\mathbb R^r)\). On such an ideal,
external product is an isomorphism by the fixed Bott inverse cycles,
with their coefficient tensor extensions, and scalar K-theory is
\(\mathbb Z\) in parity \(r\) and zero in the other parity. Tensor the
scalar exact sequence with the graded vector space
\(K_*(A)\otimes\mathbb Q\); this is exact because it is a vector space
over a field. Tensoring the coefficient sequence with \(\mathbb Q\) is
also exact. The diagrams commute by the boundary calculation just given.
Induction and the explicit five-term exact-sequence chase prove an
isomorphism on each skeleton: for surjectivity, lift the boundary image
using the outer isomorphisms, subtract that lift, and lift the remaining
kernel; for injectivity, its image in the next term is zero by that
term's injection, so lift from the preceding term, correct the lift by
the term before it, and use its isomorphism to obtain zero. This is the
standard five-lemma argument on a five-term segment, with its outside
maps invertible. It proves both parities in (R2). The same skeletal
exact sequences, with finite free relative groups, prove finite
generation of scalar K-groups. No general Künneth or UCT theorem is a
premise. \(\square\)

\textbf{Lemma A.10c (tracial character on a finite polyhedron).} Let
\(P\) be a finite polyhedron and \(\tau:A\to\mathbb C\) a continuous
positive trace, with \(\tau(1)=1\). There is a natural real even
character
\(\operatorname{ch}_\tau:K_0(C(P)\otimes A)\to H^{\rm even}(P;\mathbb R)\).
For a scalar bundle \(E\) and a projective \(A\)-module \(V\), \[
 \operatorname{ch}_\tau(E\boxtimes V)
       =\operatorname{ch}(E)\tau_*[V].
 \tag{R3}
\] It vanishes on every odd-by-odd external product in (R2). The
ordinary character \[
 \operatorname{ch}:K_0(C(P))\otimes\mathbb R
                 \longrightarrow H^{\rm even}(P;\mathbb R)
 \tag{R4}
\] is an isomorphism. A flat bundle with fiber \(A\) has character
\(1\).

\textbf{Proof: forms, including their cohomology.} Use differential
forms smooth on each closed simplex, extending smoothly a little beyond
it, with compatible restrictions on its faces; refinements do not change
their cohomology. The open stars of vertices form a finite cover. An
intersection of stars is nonempty exactly when those vertices span a
face, and it contracts by the affine homotopy to the barycenter of that
face. On each containing simplex the ordinary homotopy formula is \[
 H\omega=\int_0^1\iota_{\partial_t}h^*\omega\,dt,\qquad
                   dH+Hd=h_1^*-h_0^* .
 \tag{R5}
\] These identities agree on faces, proving the Poincaré lemma on every
intersection. A piecewise smooth partition subordinate to the stars is
obtained by applying a smooth bump, zero for \(t\leq\epsilon\), to their
barycentric coordinates and normalizing the sum, where
\(0<\epsilon<1/(\dim P+1)\). Supports are inside the stars. It contracts
the augmented horizontal rows of the Čech-form double complex:
\(s(\omega)_{i_0\ldots i_{r-1}}=\sum_j\rho_j\omega_{j i_0\ldots i_{r-1}}\),
with \(\delta s+s\delta=1\). The vertical Poincaré lemma and this row
contraction identify form cohomology with the simplicial cohomology of
the nerve, which is the chosen triangulation of \(P\).

For completeness this is its singular cohomology as well, by the actual
subdivision and prism proof of CF.0: use the subcomplex of singular
chains lying in one of the stars; subdivision makes its inclusion a
chain homotopy equivalence. The augmented Čech resolution of those
chains is exact, since for each small simplex the set of containing
cover indices is a nonempty simplex, contracted by adding one chosen
index. Its intersections have only degree-zero cohomology by the same
affine contractions. Thus its cohomology is the same nerve cohomology.
This proves the precise forms-to-singular comparison used here, only for
finite polyhedra and their finite mapping cones.

\textbf{Proof: the character and relative construction.} Every class is
a difference of finite matrix projections by the actual scalar picture,
\href{KT-KK-07.html\#2-the-even-scalar-fredholm-picture}{Lesson 07,
Theorem 2.2}. A continuous projection can be made piecewise smooth
without changing its class. Approximate its finitely many matrix entries
uniformly by finite sums of piecewise polynomial scalar functions times
elements of \(A\); barycentric partitions on a sufficiently fine finite
subdivision provide the approximation. Symmetrize, and retract by the
spectral cut around \(1\), whose fixed gap makes this a piecewise smooth
projection. The close-projection polar intertwiner identifies its module
with the original one. The same construction on the finite triangulated
prism \(P\times[0,1]\) proves invariance under continuous homotopies.
For naturality under a continuous map between finite polyhedra, a
sufficiently fine source subdivision has the image of each closed vertex
star inside a target open star, by the Lebesgue-number argument on the
finite target star cover. Assign such a target vertex to each source
vertex. For each source simplex the assigned vertices span a target
simplex, since their stars have a common image point; affine
interpolation gives a piecewise linear map. The straight carrier
homotopy within those target simplices joins it to the original map.
Pullback of forms is natural for the piecewise linear map and the
projection-path argument handles the homotopy, proving the needed
continuous-map naturality. The fiber module of a projection is locally
trivial as a projective \(A\)-module: close projections are intertwined
by the same explicit polar unitary. This supplies the bundle
interpretation without an assumed generalized bundle theorem.

For a piecewise smooth connection \(\nabla\), with curvature \(R\),
define \[
 \operatorname{ch}_\tau(\nabla)
       =\sum_{m\geq0}\frac{1}{(-2\pi i)^m m!}
                          \operatorname{Tr}_\tau(R^m).
 \tag{R6}
\] The sum terminates by dimension; degree zero is the trace of the
fiber projection. Here \(\operatorname{Tr}_\tau\) means the ordinary
matrix trace followed by \(\tau\), restricted to the projection corner.
Connections exist by the projected connection \(p\,d\) and partitions.
In a trivialization \(R=d\omega+\omega^2\), direct expansion gives
\(dR+\omega R-R\omega=0\). The graded trace of a commutator is zero, so
(R6) is closed. For a path of connections, \[
 \partial_t\operatorname{Tr}_\tau(R_t^m)
       =m\,d\operatorname{Tr}_\tau(\dot\nabla_t R_t^{m-1})
 \tag{R7}
\] by the same expansion, cyclicity and the just-proved curvature
identity. Thus changing connection changes the character by an exact
form. Additivity is the block-sum formula, and unitary module
isomorphisms conjugate the curvature without changing its trace.
Equations (R5)--(R7) prove homotopy invariance, giving the asserted map
on K-classes. For Hermitian connections its components are real. For a
flat bundle the curvature is zero; its fiber trace is one, giving
character \(1\) in every path component.

These statements have a compact-support and relative version, which will
also be needed. A relative projection pair may be made equal, with its
specified comparison, on a collar of the collapsed subcomplex or near
infinity. Choose the two connections to agree through that comparison
there; extend them using the affine space of connections and a
partition. Their difference character is compactly supported (or
relative), and (R7) is a compactly supported transgression for any
change respecting that comparison. The finite mapping cone of a
subcomplex is again a finite polyhedron, so (R5) applies there;
collapsing its contractible cone gives the relative group by CF.2 and
the actual excision of Lesson 14. On a line, a compact-support
projection pair can be made equal outside a compact interval: at
infinity it is close to its constant limiting projection, and the
close-projection unitary is joined to \(1\) and tapered off there. The
same argument works on \(P\times\mathbb R\). Thus this construction
applies to the projection definitions of the suspension K-groups as
well.

Product connections for a scalar bundle and an \(A\)-module bundle have
curvature \(R_E\otimes1+1\otimes R_V\); the summands commute as even
forms. Expanding the exponential and taking the trace proves (R3), and
proves the corresponding relative and compact-support products. For the
planar projection \(q\) of RK.6, the unit frame
\(v=(1,z)/\sqrt{1+|z|^2}\), in the coordinate \(z=re^{i\theta}\), gives
\[
 \omega=v^*dv=\frac{i r^2}{1+r^2}\,d\theta,\qquad
 R=d\omega=\frac{2i r}{(1+r^2)^2}\,dr\wedge d\theta.
\] Consequently \[
 \frac{-1}{2\pi i}\int_{\mathbb R^2}R
   =-\int_0^\infty\frac{2r}{(1+r^2)^2}\,dr=-1.
\] This is the character of \([q]-[P]\): the constant projection has
zero positive-degree curvature, while the rank terms cancel. The same
normalization follows from Stokes on the two trivializing discs, with
transition of the tautological line of degree \(-1\). It agrees with
CF.5b. Changing the sign of a Bott generator changes this to \(+1\).
Accordingly fiber integration of the compact-support character of an
exterior Bott product is \(c_b\) times the original character, where
\(c_b=\pm1\) is the explicitly chosen Bott sign. It is an invertible
scalar, retained in the following suspension comparisons.

For \(u\in K_1(C(P))\) and \(v\in K_1(A)\), their suspension
representatives lie in \(K_0^c(P\times\mathbb R)\) and
\(K_0^c(\mathbb R;A)\). For the latter relative bundle, (R6) is
identically zero: curvature two-forms on a line vanish, while its
degree-zero trace difference is locally constant and zero near infinity,
hence zero on the entire line. The compact-support product character is
therefore zero on \(P\times\mathbb R^2\). The inverse Bott comparison
and its nonzero \(c_b\) then give
\(\operatorname{ch}_\tau(u\boxtimes v)=0\). This proves the odd-by-odd
assertion with the actual suspension representatives, irrespective of
their product's fixed Clifford sign.

\textbf{Proof of (R4), without a rational comparison assertion on
arbitrary compact spaces.} Use the preceding scalar relative character
and define the odd character by the suspension representative and fiber
integration over \(\mathbb R\). They are natural and compatible with the
finite-pair exact sequences: the connecting maps are induced by the
collapse maps of the finite mapping cones, as in Lesson 14's
mapping-cone proof and CF.2; the character respects those maps and the
suspension construction. When closing the two-periodic sequence, retain
the just-computed factor \(c_b\). Multiplying the corresponding closing
cohomology arrow by this same \(c_b\) keeps it exact and makes the
diagram commute. This states the sign rather than treating a positive
course Bott element as an unspecified positive cohomology generator.

On an \(r\)-cell the character is an isomorphism over \(\mathbb R\): in
even dimension its generator is an ordered exterior product of planar
Bott generators, with character integral a product of the factors
\(c_b=\pm1\), by the compact-support product formula and Fubini. In odd
dimension suspension gives the same nonzero normalized value.
Degree-zero cells give rank. Relative groups are direct sums of these
cell groups. The scalar K-groups are finitely generated by A.10b, so
tensoring their exact sequences with \(\mathbb R\) preserves exactness.
The finite-pair diagrams just proved, induction on the skeleta and the
same five-term chase prove an isomorphism on the whole complex in both
parities. The even one is (R4). This proves precisely the injectivity
used below; it imports neither a general manifold index formula nor a
comparison theorem on an arbitrary compact Hausdorff base. \(\square\)

\textbf{Proposition A.10d (the all-cycle tracial pairing identity).} For
a flat rank-one \(A\)-module bundle \(L\) over a finite polyhedron
\(P\), with the normalized trace just specified, and every
\(y\in KK_0(C(P),\mathbb C)\), \[
 \tau_*\!\left([L]\otimes_{C(P)\otimes A}
                                 (y\boxtimes1_A)\right)
             =\langle[1_P],y\rangle\in\mathbb Z .
 \tag{R8}
\]

\textbf{Proof.} By (R2), rationally write
\([L]=\sum_\ell a_\ell\boxtimes v_\ell+
       \sum_m b_m\boxtimes w_m\), where
\(a_\ell\in K_0(C(P))\otimes\mathbb Q\), \(v_\ell\in K_0(A)\),
\(b_m\in K_1(C(P))\otimes\mathbb Q\) and \(w_m\in K_1(A)\), allowing
rational coefficients in the scalar factors. Equations (R3) and A.10c
give \[
 1=\operatorname{ch}_\tau(L)
       =\operatorname{ch}\!\left(
                     \sum_\ell\tau_*(v_\ell)a_\ell\right).
\] Injectivity in (R4) makes the real K-class in parentheses equal to
\([1_P]\). Kasparov exterior interchange shows that pairing the even
terms with \(y\boxtimes1_A\) gives \(\langle a_\ell,y\rangle v_\ell\).
Pairing the odd terms gives zero: their scalar factor lies in
\(KK_1(\mathbb C,\mathbb C)=0\). All rational equalities are legitimate
under \(\tau_*\), since a homomorphism into \(\mathbb R\) kills torsion
and extends uniquely over \(\mathbb Q\). Applying the real extension of
the integer-valued scalar pairing to the equality of real K-classes
proves (R8). Its right side is the actual scalar Fredholm index from
Lesson 07, hence is an integer. No assumption on the nature of the
representative of \(y\) was made. \(\square\)

\textbf{Theorem A.10 (torsion-free assembly trace integrality).} For
every countable torsion-free discrete group \(\Gamma\), \[
 \tau_*(\mu_r(x))\in\mathbb Z
       \quad\text{for every }x\in RK_0^\Gamma(E\Gamma;\mathbb C).
 \tag{R9}
\] Consequently surjectivity of reduced assembly implies that the only
projections and idempotents in the unamplified \(C_r^*(\Gamma)\) are
\(0,1\).

\textbf{Proof.} A class is represented at a proper \(\Gamma\)-compact
stage \(X\). Its action is free by the last assertion of A.7. Write
\(y=\Theta_X(x)\) on the compact quotient \(Y\), using (Q5). A.10a gives
a proper equivariant map to the covering \(Z\to P\) of a finite
polyhedron, with quotient map \(f_0:Y\to P\). The quotient
identification is natural for this map, as can be checked without a
general Morita-naturality assertion. Choose a cutoff on \(Z\) and pull
it back to \(X\); it has compact support by properness of the map and is
normalized by equivariance, as in A.3. Then \(f^*p_Z=p_X\). The Morita
source modules in (Q5) are \(p_ZD_Z\) and \(p_XD_X\), and extending the
first through \(f^*\rtimes\Gamma\) gives the second. Their left
invariant-function action is pulled back by \(f_0^*\). The kernel
unitary in A.8b identifies these modules with the canonical
cutoff-independent quotient modules. Associativity therefore gives
\(\Theta_Z(f_*x)=f_{0,*}\Theta_X(x)\) on every cycle class. Assembly
naturality and (Q6) now give \[
 \mu_r(x)=
 [L_P]\otimes_{C(P)\otimes C_r^*(\Gamma)}
                   ((f_{0,*}y)\boxtimes1).
 \tag{R10}
\] Apply (R8), with the normalized trace proved in A.6. Its integer
value is \(\langle1_P,f_{0,*}y\rangle=\langle f_0^*1_P,y\rangle
=\langle1_Y,y\rangle\). Here \(f_0\) is a map between compact quotients
and its pullback is unital, so the last equality is exact. Thus (R9)
holds for every analytic cycle at every stage and respects the direct
limit by A.7. Proposition A.6 gives the stated projection and idempotent
consequence under surjectivity. The same proof gives the trace formula
for full assembly followed by the regular quotient.

For an even-dimensional covering Dirac, M.4 and (R8)--(R10) also give
\(\tau_*\operatorname{Ind}_{C_r^*(\Gamma)}(D_{\mathcal L})
=\operatorname{index}(D_M)\), including finite-rank twists. This
statement does not need a triangulation or generation theorem for
manifolds: the finite-complex reduction applies to its covering, and its
quotient scalar class pairs with \(1\) by the actual Fredholm picture.
No identification with the von Neumann dimensions of lifted kernels is a
premise of this proof. \(\square\)

\textbf{Primary free source and exact scope.} Thomas Schick,
\emph{\(L^2\)-index, KK-theory, and connections}, arXiv:math/0306171,
Sections 5--6, Lemma 5.2, Theorem 5.4 and Proposition 6.7 give the
tracial-connection method and the vanishing of the odd-by-odd
contribution. Its externally quoted general Künneth theorem is not a
proof provider here: A.10b proves the finite-complex rational form
actually used. A.10c also proves the finite-polyhedron form/cohomology,
relative-character and rational comparison steps, and A.10a supplies the
reduction of every analytic proper stage. Therefore A.10d--A.10 apply to
all analytic cycles; they are not an elliptic-only argument silently
extended by a missing generation theorem. The separate von
Neumann-kernel definition of the \(L^2\)-index is not a premise of (R8).

\subsection{7. The conjecture and its mathematical
scope}\label{the-conjecture-and-its-mathematical-scope}

For a given classifying space \(\mathcal E G\) for proper actions, the
\textbf{Baum--Connes conjecture} asserts that \[
\mu_r:RK_i^G(\mathcal E G;\mathbb C)\longrightarrow K_i(C_r^*(G))
\] is an isomorphism in both degrees. Its \textbf{coefficient version}
asserts that (A16) is an isomorphism for every separable \(G\)-algebra
\(A\). These are assertions about the reduced crossed product. Formula
(A14) does not assert that the full assembly map or the coefficient
quotient is always an isomorphism.

The following are statements of known results, not premises of the
proofs in Sections 1--6. The freely accessible survey by Gomez Aparicio,
Julg and Valette, Section 1.5, records the coefficient theorem for
groups with the Haagerup property and for hyperbolic groups. Here the
Haagerup property means the existence of a metrically proper continuous
affine isometric action on a Hilbert space; for a finitely generated
discrete group, hyperbolicity means that some Cayley graph has uniformly
thin geodesic triangles. Their Section 2.4, Theorem 2.6, records the
scalar-coefficient Connes--Kasparov theorem for almost connected groups,
meaning that the group of components is compact. The survey also
describes Lafforgue's scalar-coefficient results for semisimple groups
and discrete subgroups satisfying his specified rapid-decay and
geometric hypotheses, in Section 6.1.

The coefficient formulation has counterexamples: the same survey's
Section 7.2 distinguishes groupoid counterexamples from the
transformation-groupoid examples giving group counterexamples with a
suitable coefficient algebra. Valette's freely available 2026 course
states the coefficient formulation and its limitation in Section 7.3.2.
These counterexamples are not claims that scalar assembly fails for
every group in question. No proof of these general status statements is
given here.

Two further assigned statements are recorded without use in the proofs.
The \textbf{strong Novikov assertion} is rational injectivity of the
reduced assembly map from \(K_*(B\Gamma)\); its higher-signature
implication asserts that the numbers
\(\langle L(M)\smile f^*\alpha,[M]\rangle\), for \(f:M\to B\Gamma\) and
\(\alpha\in H^*(B\Gamma;\mathbb Q)\), are invariant under oriented
homotopy equivalence of the manifold with its reference map. This
implication and its signature-class interpretation are stated here. For
a free group \(F_r\), scalar reduced assembly is an isomorphism; the
Pimsner--Voiculescu computation gives \(K_0(C_r^*(F_r))=\mathbb Z[1]\)
and \(K_1(C_r^*(F_r))=\mathbb Z^r\), generated by its free-generator
unitaries. These are survey statements in Gomez Aparicio--Julg--Valette,
Sections 2.5 and 3.2, and Valette, Sections 7.4.1 and 10.2, not
prerequisites for the lattice, trace or finite-group proofs.

For a countable torsion-free discrete group \(\Gamma\), Theorem A.10
proves the \textbf{assembly trace integrality statement} \[
\tau_*\!\left(\operatorname{im}
 \left(\mu_r:RK_0^\Gamma(\mathcal E\Gamma)\to
                         K_0(C_r^*(\Gamma))\right)\right)
 \subset\mathbb Z.
\tag{A29}
\] The proof is (R1)--(R10): it reduces every free cocompact analytic
stage to a finite complex and proves the flat-bundle tracial pairing
there. It uses neither an unproved geometric generation theorem nor the
separate von Neumann-kernel formulation of the covering index. If
reduced assembly is surjective, (A29) and Proposition A.6 prove the
torsion-free Kadison--Kaplansky projection and idempotent conclusion.

Theorem A.9 proves the lattice assembly isomorphism: \[
\mu_r:KK_i^{\mathbb Z^n}(C_0(\mathbb R^n),\mathbb C)
       \longrightarrow K_i(C_r^*(\mathbb Z^n))
\quad\hbox{is an isomorphism.}
\tag{A30}
\] The actual cutoff/descent map is identified with ordered iterated
Connes--Thom by (T10)--(T11), including every analytic class through the
proved torus duality (T4). The holonomy-compatible Green module (T6)
includes the target inversion required by the specified Mishchenko
convention. The finite-group assembly isomorphism is Theorem A.5.

\subsection{8. Exercises and solutions}\label{exercises-and-solutions}

\textbf{8.1.} Compute the cutoff projection for translation of
\(\mathbb Z\) on \(\mathbb R\).

\textbf{Solution.} Use \(h(x)=\max(1-|x|,0)\). At \(x=k+t\),
\(0\leq t\leq1\), its two nonzero translates have values \(1-t,t\), so
their sum is one. Thus \(c=\sqrt h\) is a cutoff and
\(p(n,x)=\sqrt{h(x)h(x-n)}\). The values vanish outside the
adjacent-interval overlaps for \(n=\pm1\), and outside \(n=0,\pm1\). The
convolution proof (A7) shows it is a projection. Its class is
independent of the chosen normalized cutoff by the explicit path and
partial-isometry proof of Proposition A.2.

\textbf{8.2.} Identify finite-group assembly with Green--Julg.

\textbf{Solution.} The cutoff for the point is the averaging projection
\(p_G\). Its range in the crossed regular module consists of the tuples
\(\zeta_f(h)(t)=\alpha_{t^{-1}}f(th)\). Equation (A22) gives the exact
crossed-product inner product, and (A23) is the resulting unitary.
Equation (A24) identifies invariant compact operators and their
multipliers with the same operators in the proved Green--Julg
correspondence. Compression of an invariant cycle operator therefore
gives that correspondence on every cycle and homotopy, as proved in
Theorem A.5.

\textbf{8.3.} Deduce the idempotent conclusion from surjective assembly
and integral trace on its image.

\textbf{Solution.} Surjectivity puts every \(K_0\)-class in the
integral-trace image. A projection in the unamplified reduced algebra
has trace in \([0,1]\), hence trace zero or one. The faithful trace
proof preceding Proposition A.6 gives projection zero or identity. For
an idempotent \(e\), the explicitly constructed projection
\(p=ee^*(1-(e-e^*)^2)^{-1}\) has \(pe=e\) and \(ep=p\). Either possible
value of \(p\) therefore forces the same value of \(e\). Theorem A.10
supplies the integral-trace hypothesis for every countable torsion-free
group; with surjective reduced assembly, this solves the assigned
torsion-free consequence.

\textbf{8.4.} Prove the \(\mathbb Z^2\) assembly isomorphism using two
Connes--Thom maps.

\textbf{Solution.} Cross the second coordinate and then the first, using
the two positive-in-both-parities Connes--Thom isomorphisms. Equation
(T12) identifies their composition, followed by the specified Green
equivalence and a minus sign, with assembly under the quotient
identification (Q5) and the integral torus duality (T4). Each factor is
invertible, proving the actual assembly map is an isomorphism in both
degrees. The derivative matrices and the target coordinate inversion
that produce the minus sign are displayed immediately after (T12).

\subsection{What this lesson does not
prove}\label{what-this-lesson-does-not-prove}

The proved results are cutoff, cycle and model independence;
full/reduced assembly and its space/coefficient naturality; finite-group
and lattice isomorphisms; the covering Dirac and all-cycle Mishchenko
comparisons; and integral trace on the entire torsion-free assembly
image, with the projection/idempotent consequence under surjectivity.
The general Baum--Connes conjecture and the group/status,
higher-signature and free-group results in Section 7 are stated and
unused. This lesson does not prove those survey theorems or identify the
numerical tracial index with the separate von Neumann-kernel definition
of the L²-index.

\subsection{Freely accessible reading}\label{freely-accessible-reading}

\begin{itemize}
\item
  Markus Land, \href{https://arxiv.org/abs/1306.5657}{\emph{The
  Analytical Assembly Map and Index Theory}}, Sections 2--4, for
  assembly, Mishchenko bundles and their comparison.
\item
  Alain Valette, \href{https://arxiv.org/abs/2610.01802}{\emph{The
  Baum--Connes conjecture: a concise course}}, version 1, 1 October
  2026, Sections 4.1, 7.2--7.4 and 8, for the formulations and
  covering-index/Dirac context.
\item
  Maria Paula Gomez Aparicio, Pierre Julg and Alain Valette,
  \href{https://arxiv.org/abs/1905.10081}{\emph{The Baum--Connes
  conjecture: an extended survey}}, Section 1.5, Section 2.4, Section
  6.1 and Section 7.2, for the explicitly stated status results. Its
  2019 list of open cases is not asserted as a current list of open
  problems.
\item
  Wolfgang Lück, \href{https://arxiv.org/abs/math/0312378}{\emph{Survey
  on classifying spaces for families of subgroups}}, Definitions 1.8 and
  2.1 and Theorem 2.5, for the universal proper/numerable distinction;
  the used model comparison is proved in (U1)--(U8).
\item
  Thomas Schick,
  \href{https://arxiv.org/abs/math/0306171}{\emph{L²-index, KK-theory,
  and connections}}, Sections 5--6, for the tracial connection method;
  the finite-complex and all-cycle steps actually used are proved in
  (R1)--(R10).
\end{itemize}

\end{document}