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\section{The longitudinal index theorem and assembly for
foliations}\label{the-longitudinal-index-theorem-and-assembly-for-foliations}

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

Let \(V\) be a closed, Hausdorff, second-countable smooth manifold and
let \(F\subset TV\) be an integrable smooth subbundle of constant rank
\(p\). Write \[
 G=\operatorname{Hol}(V,F),\qquad A=C_r^*(G),\qquad
 B_F=C_0(F^*),\qquad C_F=\Gamma(V,\mathrm{Cl}^+(F)).
 \tag{0.1}
\] The arrow manifold of \(G\) can be non-Hausdorff. In that case a
smooth compact kernel means a finite sum of kernels supported compactly
inside Hausdorff arrow charts, extended by zero; equal presentations are
identified by their operators on the holonomy covers. The unit manifold
and the individual covers are Hausdorff. Kernels carry the endpoint
half-densities, or equivalently one fixed smooth positive leaf density.
Throughout, \(K_c^j(F^*)=K_j(B_F)\); compact support in the fibre is
part of this notation.

There are two questions. The index theorem identifies the class of a
longitudinal elliptic symbol with a topological construction using an
embedding of \(V\). Assembly asks how much of \(K_*(A)\) is described by
geometric cycles over the leaf space. Equality of two index
constructions does not assert that either map, or assembly, is an
isomorphism.

We use the proved product and Clifford Thom equivalences in
\href{KT-KK-09.html}{Connections and the existence of the Kasparov
product}, \href{KT-KK-10.html}{Homotopy, associativity, the index
pairing and KK-equivalence}, and \href{KT-KK-13.html}{Thom isomorphisms
and K-orientations in KK}. For ordinary manifolds the full wrong-way
theorem in \href{KT-KK-16.html}{Wrong-way maps for K-oriented maps},
Theorems 6.1, 7.3 and 8.1, supplies composition, graph-factorization
independence and genuine interval homotopies. A map to a leaf space
additionally needs a completed groupoid correspondence. Those
correspondences and their geometric-cycle relations are proved in
\href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-6}{K-theory
of the leaf space}, Section 6; we specify exactly which ones below.

\subsection{1. The analytic symbol-class
map}\label{the-analytic-symbol-class-map}

For a finite Hermitian bundle \(E\to V\), choose an isometric embedding
in \(V\times\mathbb C^k\), with projection \(e_E\in M_k(C(V))\). Its
module is \[
 \mathscr E_E=e_E A^k,\qquad
 \mathcal K_A(\mathscr E_E,\mathscr E_H)=e_HM_{l,k}(A)e_E.
 \tag{1.1}
\] Its regular tensor fibre at \(x\) is \(L^2(G_x;r^*E)\). The
projection is a multiplier, and the formula for the compact maps follows
by multiplying column kernels and their adjoints and using an
approximate identity of \(A\). These modules are countably generated: a
countable arrow atlas with countable compact-chart approximations gives
a countable dense family in \(A\), and the finite column module inherits
it. Different finite bundle embeddings give isomorphic modules; the
rotation of two isometric embeddings gives their stabilized comparison.

\textbf{Proposition 1.1 (analytic index).} There is a homomorphism \[
 \operatorname{Ind}_a:K_c^0(F^*)\longrightarrow K_0(A)
 \tag{1.2}
\] which takes the relative principal symbol of a longitudinal classical
elliptic operator to its module Fredholm index, with sign kernel minus
cokernel. The odd version is obtained with the same right-Clifford
suspension convention.

\textbf{Proof.} The precise earlier proof is
\href{../NCG-FOLIATIONS/hilbert-modules-and-fields-on-the-leaf-space.html\#section-6}{Hilbert
modules and fields on the leaf space}, Lemma 6.5 and Theorem 6.6,
including formulas (11)--(13). Its analytic estimates are
\href{../NCG-FOLIATIONS/the-index-theorem-for-measured-foliations.html\#section-6}{The
index theorem for measured foliations}, Lemmas 6.1--6.2 and Proposition
6.21. We recall the actual mechanism to fix the hypotheses.

In a finite protected plaque atlas, quantization uses \[
 (2\pi)^{-p}\int e^{i(t-t')\cdot\xi}a(t,\xi,u)v(t')\,dt'\,d\xi .
\] For compact \(t\)-support, its Fourier kernel satisfies
\(\|\widehat a(\eta-\xi,\xi,u)\|\le C_N\langle\eta-\xi\rangle^{-N}\langle\xi\rangle^m\),
uniformly in the transverse parameter \(u\). Schur's estimate gives the
order-zero bound from finitely many seminorms. Each lifted arrow chart
is a sum of mutually orthogonal plaque-sheet operators, so this bound
remains uniform over all covers. For \(m<0\), cutting the frequency at
radius \(R\) gives smooth compact arrow kernels, and the omitted
order-zero seminorms are \(O(R^m)\). Thus negative order means
membership in the matrix corner of \(A\), and hence module compactness.
Ordinary compactness on an infinite holonomy cover is neither needed nor
asserted.

Composition and adjoint expansions are finite Taylor expansions in
\(t'-t\), with integration by parts in frequency; their first remainders
have order minus one. Coordinate changes are handled by the invertible
averaged Jacobian near the diagonal. These calculations imply that an
order-zero elliptic symbol \(u\) and a quantization of \(u^{-1}\) give
adjointable operators \(T:X\to Y\), \(Q:Y\to X\) with \(QT-1,TQ-1\)
compact. The explicit invertible lift is \[
 W=\begin{pmatrix}T(2-QT)&TQ-1\\1-QT&Q\end{pmatrix}:
 X\oplus Y\longrightarrow Y\oplus X.
\] It is the product of three elementary triangular matrices and the
signed flip. If \(e_X,e_Y\) are the first-summand projections,
\(We_XW^{-1}-e_Y\) is compact. Add the finite bundle complements to make
the reference a constant matrix projection. Its difference class is the
index in \(K_0(A)\). Varying \(Q\) along its affine path or varying
\(T\) through a norm-continuous Fredholm family gives an idempotent
homotopy. For finite-dimensional maps this class is \(\dim X-\dim Y\).

A relative class on the disk/sphere pair of \(F^*\) consists of two
bundles over the disk and an isomorphism on the sphere. Radial
contraction and the polar comparison of two projections at distance less
than one identify the two disk bundles with pullbacks from \(V\). Smooth
approximation on the compact sphere retains invertibility. Quantize that
smooth sphere symbol, extended homogeneously above a fixed radius and
smoothed near zero. Uniform approximation and homotopies give
norm-continuous families by the same finite-seminorm estimate. If the
sphere isomorphism extends invertibly to the disk, contraction gives an
invertible bundle multiplier and index zero. Addition is direct sum.
Hence every defining relative K-theory relation has the same image. This
proves (1.2) for all compact relative classes, rather than only
differential symbols of a fixed order. Nonzero orders can first be
reduced in the proved longitudinal Sobolev calculus; the construction
itself only needs classical order-zero representatives.

For non-Hausdorff arrows, the singular components lie in the Hausdorff
near-unit disk chart; smooth components retain their finite chart
presentations. Composition, frequency truncation and all the bounds
above apply component by component. This is precisely the content of
Proposition 6.21. Suspending the same completed module and boundary
calculation gives the odd assertion. When \(p=0\), the cosphere is empty
and the finite bundle difference is treated directly, as in Section 3.
\(\square\)

\subsection{\texorpdfstring{2. A topological index without orienting
\(F\)}{2. A topological index without orienting F}}\label{a-topological-index-without-orienting-f}

Fix a smooth embedding \(i:V\hookrightarrow\mathbb R^N\), with \(N\)
even. Such an embedding can be constructed from a finite coordinate
cover: if smooth chart-supported weights \(\rho_a\) sum to one, the map
with components \(\rho_a\) and \(\rho_ax_a\) is injective and an
immersion. A positive weight recovers a chart coordinate and detects a
nonzero derivative. Compactness makes it an embedding. Append zero
coordinates to make \(N\) even. Put \[
 \nu_i=(di(F))^\perp\subset V\times\mathbb R^N,\qquad q=N-p.
 \tag{2.1}
\] Neither \(F\) nor \(\nu_i\) is assumed spin\(^{c}\). Choose a metric
on \(F\); let \(e_i:F\to\mathbb R^N\) be the isometric injection
obtained from \(di|_F\) by its positive Gram inverse square root. Its
image has the same complement \(\nu_i\). Metric changes use the
positive-matrix homotopy of the Thom theorem.

Set \(\epsilon_r=(-1)^{r(r-1)/2}\). Let \(S_N\) be the constant
irreducible complex ambient spinor with positive Clifford operators
\(\gamma(v)^2=|v|^2\) and grading \(\Gamma_N\). This retains the whole
trivial ambient spin structure.

\textbf{Lemma 2.1 (complement Thom equivalence).} The finite bundle
cycle \[
 z_i=\left(C_0(\nu_i,S_N),\quad
 v\longmapsto-\gamma(e_i v),\quad
 \frac{\gamma(\eta)}{\sqrt{1+|\eta|^2}},\quad
 \epsilon_p\Gamma_N\right)
 \in KK(C_F,C_0(\nu_i))
 \tag{2.2}
\] is a KK-equivalence. If \(x_F\in KK(C_F,B_F)\) is the Clifford Thom
equivalence of the Thom lesson, Theorem 4.2, with inverse \(y_F\), then
\[
 \Theta_i=y_F\otimes_{C_F}z_i
 \in KK(B_F,C_0(\nu_i))
 \tag{2.3}
\] is a KK-equivalence.

\textbf{Proof.} The source action of a Clifford generator is odd. It
anticommutes with the odd normal position operator, so its graded
commutator is zero. The square defect is the scalar \(-1/(1+|\eta|^2)\),
vanishing at fibre infinity uniformly over the compact base.
Multiplication by a source section makes it a compact finite-bundle
endomorphism. The action is an essential graded *-representation.

Apply the standard Clifford-coefficient Thom element \(\beta_{\nu_i}\)
from Sections 2--3 of the Thom lesson, with coefficient \(C_F\). The
joint positive Clifford algebra of the orthogonal sum
\(F\oplus\nu_i\cong\underline{\mathbb R^N}\) is represented on \(S_N\)
by \((-\gamma(e_iv),\gamma(\eta))\). It is the full matrix algebra: the
Clifford-word basis has dimension \(2^N\), and its spinor matrix
representation has that same dimension and no kernel, as follows from
the Clifford anticommutation relations and the orthogonality of its
words for the matrix trace. It glues using the actual ambient Clifford
action. The grading \(\epsilon_p\Gamma_N\) implements the grading of
that matrix algebra; reversing a finite Morita module's grading still
gives an invertible Morita module. Compose this full matrix Morita
equivalence with the Thom element. The result is exactly (2.2). The
inverse is its finite conjugate module followed by \(\delta_{\nu_i}\).
The two Thom products in Sections 2--3 and the two finite evaluation
maps \(s\otimes t^*\mapsto\theta_{s,t}\),
\(s^*\otimes t\mapsto\langle s,t\rangle\), prove both inverse
identities. Thus no separate spin\(^{c}\) choice for \(F\) or \(\nu_i\)
has entered the argument. Formula (2.3) and the two identities for
\(x_F,y_F\) prove its assertion. \(\square\)

The minus sign and \(\epsilon_p\) in (2.2) are part of the comparison.
On the balanced module for (2.3), the inverse \(y_F\) is the vertical
differential operator \(i\epsilon_p L_e\partial_\xi\) with grading
\(\epsilon_p\Gamma_{\mathrm{Cl}}\), as proved in the
\href{KT-KK-15.html\#5-the-clifford-inverse-and-the-two-cotangent-conventions}{closed-index
lesson, Lemma CI.5}. Evaluation on \(S_N\) gives, after an even
conjugation removing a common operator sign, \[
 D_{\Theta_i}=-i\gamma(e_i e_j)\partial_{\xi_j}+\gamma(\eta),
 \qquad\Gamma_{\Theta_i}=\Gamma_N,
 \qquad D_{\Theta_i}^2=-\Delta_\xi+|\eta|^2.
 \tag{2.4}
\] The source \(B_F\) multiplies the \(\xi\) variable. Fourier transform
in each vector fibre constructs the regular resolvents. With a compact
\(\xi\)-cutoff, bounded \(\eta\)-regions give Rellich compactness, and
the resolvent norm on the remaining region is at most
\((1+|\eta|^2)^{-1/2}\). Finite base charts therefore give
source-localized module compactness. The smooth vertical core is dense
in both graph domains. The product connection identity follows by
differentiating a smooth creation section; its error is its compactly
supported derivative. The graded anticommutator with the old vertical
operator is nonnegative. These are the connection, domain and positivity
checks of that inverse product, now with the finite ambient Morita
module. They also prove (2.4) as a representative, although
invertibility of (2.3) already follows from the two proved factors.

\textbf{Lemma 2.2 (a transverse tube and its reduced correspondence).} A
sufficiently small disk neighbourhood \(N_i\subset\nu_i\) of the zero
section has the transverse embedding \[
 \phi_i(x,\eta)=(x,i(x)+\eta)
 \quad\hbox{in }(V\times\mathbb R^N,F\oplus0).
 \tag{2.5}
\] It defines a class \[
 \kappa_i\in KK(C_0(\nu_i),A\otimes C_0(\mathbb R^N)).
 \tag{2.6}
\] The class uses the intrinsic holonomy of a small foliated tube,
including its open-subgroupoid inclusion in \(G\times\mathbb R^N\).

\textbf{Proof.} At \((x,0)\), transversality says that \(di(F_x)\)
together with \(\nu_{i,x}\) spans \(\mathbb R^N\). This is (2.1). The
derivative remains invertible in the transverse directions on a small
disk, by compactness and continuity. The first coordinate in (2.5) makes
it injective. A positive smooth radial compression identifies the whole
normal bundle with such an open disk.

Flow a small vector in \(F_x\) leafwise from \((x,i(x)+\eta)\), keeping
the Euclidean coordinate fixed. The leafwise exponential gives a
diffeomorphism from a disk bundle over \(N_i\) to an open set
\(\Omega_i\) after shrinking both radii. Its derivative is the
tangent/transverse isomorphism at the compact zero section. If
arbitrarily small radii failed injectivity, two colliding points would
converge to the same point of that section, contradicting its local
inverse. The same argument on a slightly smaller closed disk gives
uniform control. In these coordinates the foliation consists exactly of
the connected plaque-ball fibres of a submersion \(r:\Omega_i\to N_i\).
Its intrinsic groupoid is \(H_i=\Omega_i\times_{N_i}\Omega_i\), not the
full reduction of all global holonomy arrows to \(\Omega_i\).

The completed fibre-integration module \(E_r=L^2(\Omega_i/N_i)\) has
inner product
\(\langle\xi,\zeta\rangle(n)=\int_{r^{-1}(n)}\overline\xi\zeta\). The
correspondence of \href{../KT-CP/KT-CP-16.html}{C*-algebras of
foliations and their Morita equivalences}, Propositions 16.3--16.4,
identifies \(C_r^*(H_i)\) with \(\mathcal K(E_r)\). On a compact
submersion chart the kernel of a rank-one map is
\(\xi(y)\overline{\zeta(y')}\); finite product approximations and their
uniform integral bounds prove norm density of these kernels. Positive
plaque bump functions normalized to fibre norm one show fullness over
\(C_0(N_i)\). Thus the conjugate module is an actual inverse Morita
class \(C_0(N_i)\to C_r^*(H_i)\).

Here is the reduced-norm assertion required to extend that class. For
any open subgroupoid \(H\subset G'\), a regular source fibre \(G'_x\)
splits, on the part with range in \(H^{(0)}\), into open left
\(H\)-orbits. There are at most countably many, since the fibre is
second countable. Choose \(a:x\to y\) in one orbit. The map
\(h\mapsto ha\) identifies that orbit with \(H_y\), preserves the lifted
leaf half-density, and intertwines convolution by an \(H\)-kernel with
its regular \(H_y\)-representation. On the inactive part the
representation is zero. Hence the norm of the zero-extended kernel is at
most its reduced \(H\)-norm. For \(x\in H^{(0)}\), the \(H_x\)-summand
gives the opposite bound. Zero extension is consequently an isometric
*-homomorphism \(C_r^*(H)\to C_r^*(G')\). The argument works for finite
chart kernels and so for non-Hausdorff arrows as well.

Apply this to \(H_i\subset G\times\mathbb R^N\). For the product
groupoid the regular norm is the supremum in the Euclidean coordinate of
the regular \(G\)-norm. Finite compact kernel tensor approximations are
dense; therefore its algebra is
\(C_0(\mathbb R^N,A)=A\otimes C_0(\mathbb R^N)\). Compose the conjugate
\(E_r\), the open inclusion and the radial transport of \(C_0(\nu_i)\).
This proves (2.6). The module is countably generated by countable chart
cores; its source action is compact because it is a Morita compact
action followed by tensoring a homomorphism. It is a valid zero-operator
cycle even though the open inclusion need not act essentially on the
entire target algebra. \(\square\)

\textbf{Lemma 2.3 (tube choices and actual families).} Changes of the
radial compression, the small foliated tube, the leaf density or the
metric give the same product \(\Theta_i\kappa_i\). A smooth
compact-parameter family of embeddings gives a KK homotopy of these
products with interval coefficients.

\textbf{Proof.} Two normalized tubes agree to first order on their
section. On a common smaller disk their transition has invertible
derivative joined through positive metric transport to the identity.
Radial contraction to the section and interpolation of the resulting
local sprays give a foliated tube family with the same section; the
finite compact-section inverse argument gives a common positive radius.
This is the tubular-family argument of the wrong-way lesson, Lemma 2.1
and Theorem 8.1, applied in the leafwise/transverse charts. The family
preserves the plaque submersions: its construction flows within the
fixed leaves from the varying transversal.

Construct its whole submersion module over \(C_0(\mathcal N)\), where
\(\mathcal N\to I\) is the transverse normal family. The inner product
is the fibre integral on each plaque; compact parameter charts make that
integral continuous. Compact smooth sections are dense, and their
endpoint restrictions are onto a dense endpoint core. Multiplication by
parameter cutoffs and extension in a product chart show that the
completed evaluation module is exactly \(E_{r_s}\), isometrically and
surjectively. The corresponding open subgroupoid family lies in
\(G\times\mathbb R^N\times I\), whose reduced algebra is
\(C(I,A\otimes C_0(\mathbb R^N))\) by the same regular-norm argument.
The finite spinor module, its Clifford action, its position multiplier
and its compact square defect in Lemma 2.1 are defined on the whole
parameter normal bundle. Their compact defects vanish uniformly at fibre
infinity. Its inverse vertical differential module has the uniform
resolvent estimates of (2.4). Their product with the completed interval
submersion module is therefore a Kasparov cycle over this interval
algebra, with precisely the claimed endpoint evaluations.

A common smaller tube can be used without declaring a neighbourhood
inclusion KK-invertible. On the essential open-support module
\(\overline{C_0(U)E}\), first compress the normal Thom cycle into \(U\).
Its defects and all nonmultiplication terms lie there, and zero
extension recovers the original tube cycle. Tensoring the completed
module identifies the two essential restrictions by the identity on
their dense section cores. This is the open-extension equality in
Proposition 7.1 and the interval proof of the wrong-way lesson; it also
follows from the explicit restriction unitary in
\href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-6}{K-theory
of the leaf space}, Lemma 6.0n. Positive density changes use their
square-root multiplication unitary. Positive metrics and the affine
family of connections give the finite-module transports in the Thom
theorem. These constructions prove all the choice assertions, including
their actual interval completion. \(\square\)

Let \(\beta_N\in KK(\mathbb C,C_0(\mathbb R^N))\) be the outward
positive scalar Bott element, and let \(\eta_N\) be its two-sided
inverse, with the grading of the Bott and Thom lessons. Define \[
 \mathcal T_i=\Theta_i\otimes\kappa_i\otimes(1_A\boxtimes\eta_N)
 \in KK(B_F,A),\qquad
 \operatorname{Ind}_{t,i}(a)=a\otimes_{B_F}\mathcal T_i.
 \tag{2.7}
\]

\textbf{Lemma 2.4 (even stabilization).} Appending an even Euclidean
block to \(i\) leaves \(\mathcal T_i\) unchanged.

\textbf{Proof.} For \(i'=(i,0)\) in \(\mathbb R^N\oplus\mathbb R^{2b}\),
its normal is \(\nu_i\oplus\mathbb R^{2b}\). The same ambient Clifford
module is the ordered graded tensor of the old one and the new standard
spinor. Its source minus sign is the old one, and its grading is
\(\epsilon_p\Gamma_N\widehat\otimes\Gamma_{2b}\). The position Clifford
sum is the direct-sum Thom product from the Thom lesson. If
\(\epsilon_p=-1\), its tensor formula initially has the extra sign on
the new position block from the old module grading; conjugation by
\(1\otimes\Gamma_{2b}\) removes it. This is an even global unitary
preserving the old Clifford source and total grading. Hence
\(z_{i'}=z_i\boxtimes\beta_{2b}\), and likewise
\(\Theta_{i'}=\Theta_i\boxtimes\beta_{2b}\). The normal/transverse tube
may be chosen as the old one times the new Euclidean coordinate.
Elementary compact sections in the new coordinate show that its
completed correspondence is
\(E_{\kappa_i}\widehat\otimes C_0(\mathbb R^{2b})\): the inner product
and right convolution factor, and uniform compact-support tensor
approximation is dense. Thus \(\kappa_{i'}=\kappa_i\boxtimes1\).

The inverse for the ordered ambient blocks first removes the new block
and then the old one. The proved two inverse products cancel the
adjacent \(\beta_{2b}\eta_{2b}\). This proves the assertion in KK. In
the native ordinary right-Clifford notation the wrong-way lesson inserts
\(\epsilon_r\) on both the embedding and the inverse projection; its
Lemma 1.1 and Proposition 7.2 give exactly this same cancellation. One
must not normalize one outward planar Bott factor in isolation and leave
its inverse or the composition convention unchanged. \(\square\)

\textbf{Theorem 2.5 (well-defined topological index).} The class
\(\mathcal T_i\), and hence
\(\operatorname{Ind}_t:K_c^j(F^*)\to K_j(A)\), is independent of the
even Euclidean embedding and of every choice above.

\textbf{Proof.} Given \(i_0:V\to\mathbb R^{N_0}\) and
\(i_1:V\to\mathbb R^{N_1}\), use \[
 i_\theta=(\cos\theta\,i_0,\sin\theta\,i_1),\qquad0\le\theta\le\pi/2.
\] At every parameter at least one coefficient is nonzero, so this is an
injective immersion and, by compactness, an embedding. Its smooth normal
complements, positive Gram isometries, ambient Morita modules and
foliated tubes form the genuine families of Lemma 2.3. The two endpoint
products agree. At the first endpoint Lemma 2.4 removes the extra
\(N_1\)-block. At the second endpoint exchange the two even coordinate
blocks. That real permutation preserves orientation. Its exterior spinor
unitary preserves the full Bott class; the two inverse identities then
preserve its inverse. The correspondence's Euclidean coordinate is
transported by that same proper linear diffeomorphism. Lemma 2.4 removes
the extra \(N_0\)-block. This proves equality of the two unstabilized
classes. The argument uses the full tube/factorization and projection
homotopy of the wrong-way lesson, together with the completed foliation
module of Lemma 2.2; it makes no use of the longitudinal index theorem.
Exterior suspension gives both parities. \(\square\)

\subsection{3. The theorem and the zero-dimensional
case}\label{the-theorem-and-the-zero-dimensional-case}

\textbf{Longitudinal index theorem (stated).} For every compact smooth
foliated manifold as above, the analytic and topological maps agree: \[
 \operatorname{Ind}_a=\operatorname{Ind}_t:
 K_c^j(F^*)\longrightarrow K_j(C_r^*(\operatorname{Hol}(V,F))),\qquad j=0,1.
 \tag{3.1}
\] In degree zero this includes every longitudinal classical elliptic
pseudodifferential operator between finite bundles. The odd assertion
uses suspension. No spin\(^{c}\) structure on \(F\), Hausdorffness of
the entire arrow manifold, or transverse invariant measure is required.
The free primary locators are Connes--Skandalis, \emph{The longitudinal
index theorem for foliations}, Theorem 4.14 and Remark 4.15, and Connes,
\emph{Noncommutative Geometry}, author edition, Chapter II, Section 9,
Theorem 6. The full theorem is stated here; Sections 3--4 prove its
specified special cases independently.

\textbf{Theorem 3.1 (zero foliation).} If \(F=0\), then \(A=C(V)\),
\(F^*=V\), and both maps in degree zero are \(1_{K^0(V)}\).

\textbf{Proof.} Every leaf is a point and the holonomy groupoid has only
units. Formula (1.1) is the finite bundle module \(\Gamma(E)\); every
adjointable bundle map is compact. Its Fredholm index is
\([E^+]-[E^-]\), irrespective of the chosen bundle map. The cosphere is
empty, so the relative symbol group is exactly \(K^0(V)\). This proves
the analytic assertion without a fictitious unit covector.

Now \(\nu_i=V\times\mathbb R^N\), \(\epsilon_0=1\), and
\(\Theta_i=1_{C(V)}\boxtimes\beta_N\). The transverse map is
\((x,\eta)\mapsto(x,i(x)+\eta)\); its tube has point fibres. The
correspondence is radial transport followed by extension by zero.
Translation by \(i(x)\) is joined to zero translation through
\(t i(x)\), with one completed interval module. Positive radial
compression gives the same compact relative Bott class, by the
normalized tube homotopy of the wrong-way theorem. Consequently
\(\Theta_i\kappa_i=1_{C(V)}\boxtimes\beta_N\) in the target coordinates.
Multiplication by \(1_{C(V)}\boxtimes\eta_N\) gives the unit by the
actual Bott inverse identity. Thus the topological index is identity as
well. The same suspension gives identity in odd degree. \(\square\)

\subsection{4. A foliation with one
leaf}\label{a-foliation-with-one-leaf}

Assume \(V\) is nonempty and connected and \(F=TV\). All transverse
holonomy germs have dimension zero, so \(G=V\times V\), even when \(V\)
has nontrivial fundamental group. The submersion calculation in the
foliation-algebra lesson, Propositions 16.3--16.4, gives \[
 A=\mathcal K(L^2(V)),\qquad A\sim_M\mathbb C.
 \tag{4.1}
\] We prove the topological comparison in the full Clifford-source KK
group. This is the point that permits odd and non-spin\(^{c}\)
manifolds.

Write \(\mathscr D_V\) for the intrinsic doubled-Clifford Dirac class of
the closed-index lesson, equation (CI.26): on
\(\Lambda F_{\mathbb{C}}\), its two positive Clifford actions are
\(c_j=\varepsilon_j+\iota_j\), \(f_j=i(\varepsilon_j-\iota_j)\), its
operator is \(-i\sum c_j\nabla_j\), and its grading is
\(\epsilon_p\Gamma_0\). Let \(d_{V,-}\in KK(C_0(T^*V),\mathbb C)\)
denote its cotangent Dolbeault comparison for \[
 J_-(v,\eta)=(g^{-1}\eta,-gv).
\] The closed-index lesson, Theorem CI.6, proves
\(y_F\mathscr D_V=d_{V,-}\). The opposite \(J_+\) convention differs by
the explicit determinant-line and grading correction there; it cannot be
exchanged silently.

\textbf{Theorem 4.1 (one-leaf comparison).} Under (4.1), \[
 z_i\kappa_i\eta_N=\mathscr D_V\quad\hbox{in }KK(C_F,\mathbb C),
 \qquad \mathcal T_i=d_{V,-}\quad\hbox{in }KK(B_F,\mathbb C).
 \tag{4.2}
\] Therefore \(\operatorname{Ind}_a=\operatorname{Ind}_t\) for this
foliation on every compact relative symbol class.

\textbf{Proof: the completed tube product.} In this case (2.5) is the
ordinary normal tubular map \(\tau(x,\eta)=i(x)+\eta\) after forgetting
the \(V\) coordinate. Tensor the submersion module in Lemma 2.2 with the
pair-groupoid Morita module. On compact section tensors, integration
over the plaque variable gives the fibre inner product on the open
normal tube. Its integral-kernel rank-one formulas identify the left
multiplication actions, and compact product approximations make these
tensors total. Thus it extends to an onto isometry with the essential
\(C_0(\tau(\nu_i))\)-module inside \(C_0(\mathbb R^N)\). The class
\(\kappa_i\), after this Morita equivalence, is exactly ordinary tubular
extension. This is equality of modules, not equality of fibre indices.

The scalar ambient inverse is the Euclidean derivative
\(-i\gamma_D\partial\) with grading \(\epsilon_N\Gamma_D\), in even
\(N\). This is the outward inverse grading fixed by the two Bott
products. Pull it to the tube, including the positive absolute
half-density Jacobian and the full ambient spinor transport. Normal
contraction and the complete connection-metric replacement are the local
Dirac comparison of the
\href{KT-KK-15.html\#10-tubular-extension-and-the-embedding-index}{closed-index
lesson, Lemma CI.12}; the ordinary tube class is Proposition 7.1 of the
wrong-way lesson. On the essential tube module, conjugate by
\(1\otimes\Gamma_D\) if \(\epsilon_p=-1\). This fixes the position term,
source action and total grading and removes the grading sign from the
inverse differential factor. The tensor sum is then \[
 H+O,\qquad
 H=-i\sum_{j=1}^p\alpha_j\nabla_j,
 \quad\alpha_j=\Gamma_\tau\otimes\gamma_{D,j},
 \quad O=\sum_{a=1}^q\gamma_{\tau,a}\eta_a
             -i\Gamma_\tau\otimes\gamma_{D,a}\partial_{\eta_a}.
 \tag{4.3}
\] Tangent indices refer to \(F\), normal indices to \(\nu_i\). Its
source Clifford generator is \(f_j=-\gamma_{\tau,j}\otimes1\) and its
total grading is \(\epsilon_p\epsilon_N\Gamma_\tau\Gamma_D\). These
formulas are global associated-bundle formulas; oriented frames of the
two separate bundles are unnecessary.

\textbf{Proof: domain and source checks.} Choose the metric connection
preserving the orthogonal tangent/normal splitting, and lift it to both
full ambient spinor factors. Parallel transport preserves the vertical
oscillator, including its position and derivative terms. Thus
\([\nabla_j,O]=0\), \(\{\alpha_j,O\}=0\), and \(HO+OH=0\) on the smooth
compact core. That core is invariant under the smooth source algebra
\(\Gamma^\infty\mathrm{Cl}(F)\). The graded commutator of \(H+O\) with a
source section is its bounded base covariant-derivative multiplier;
\(O\) graded-commutes with the source exactly. Compactness of the base
bounds these derivatives. A horizontal half-density correction is
bounded and preserves all source actions and kernel comparisons.

The completion has graph domain obtained from the horizontal first
derivative, the vertical derivatives and multiplication by \(|\eta|\).
The identity \(O^2=-\Delta_\eta+|\eta|^2-\sum_a K_a\), with \(K_a\)
below, and the closed horizontal elliptic estimate give the graph-norm
estimate; finite base partitions and increasing radial cutoffs make the
smooth core dense in the maximal domain. The cutoff errors are bounded
by the cutoff derivative, so the adjoint domain agrees with this
closure. The finite vertical Hermite decomposition, preserved by the
connection and by the tangent Clifford action, gives a particularly
explicit closure. Each fixed eigenspace of \(O^2\) is a finite-rank
bundle over the compact base. On it \(H+sO\) is a symmetric elliptic
horizontal operator plus a smooth bounded matrix, hence has the closed
domain of the closed-manifold domain lemma. Their orthogonal direct sum
is self-adjoint. On the level \(O^2=\lambda\), its inverse resolvent
norm is at most \((1+s^2\lambda)^{-1/2}\). Finite-level truncations are
compact horizontal resolvents, and these bounds make their remaining
norms tend to zero. Finite Hermite sums are a graph core; radial cutoffs
turn their Gaussian tails into compact normal support, with vertical
derivative error tending to zero and horizontal radial error zero
because the connection preserves the fibre norm. This gives regular
self-adjointness and compact resolvents, since the base is compact and
each horizontal eigenspace problem is elliptic. The resulting Hilbert
space is separable.

For product recognition, differentiate a smooth compact normal creation
section in the ambient derivative; the connection error is its compactly
supported derivative. The graded anticommutator with the normal position
factor is bounded below by \(-q\). The completed weighted domain just
established is contained in each creation domain. Local Rellich
estimates and the normal confining tail make these errors compact after
the resolvent factors. These are the connection/domain/positivity
conditions proved for the normal tensor sum in Lemma CI.13, and prove it
is the product in (4.2). The use of a full Clifford source adds no
unexplained scalar-source reduction: its graded commutator is the
bounded multiplier just calculated, and the same normal tail bounds
every finite source coefficient. In the local tube comparison its
coefficients are smooth bounded base matrices. Every derivative error is
bounded on the compact normal core, while normal position makes the
remaining tail tend to zero. The interval compactness argument of Lemma
CI.12 therefore applies to this source too. The pure position part
permits compression into the tube and essential support extension; no
arbitrary open restriction is declared a KK-equivalence.

\textbf{Proof: the exact global kernel.} Put \[
 K_a=-i\gamma_{\tau,a}\Gamma_\tau\otimes\gamma_{D,a}.
 \tag{4.4}
\] They are commuting self-adjoint involutions. Every nonempty product
has trace zero, because a remaining Clifford generator, or the matrix
chirality word, has trace zero in the full ambient matrix algebra. Hence
the simultaneous \(+1\) eigenspace has dimension \(2^N/2^q=2^p\). The
kernel of \(O\) is this space times \(\pi^{-q/4}e^{-|\eta|^2/2}\), and
\(O^2\ge2\) on its orthogonal complement. The tangent \(\alpha_j,f_j\)
commute with every \(K_a\) and satisfy the full doubled positive
Clifford relations.

The grading comparison must also be made. On the kernel, multiplication
of the ordered Clifford words gives \[
 \Gamma_{0,F}:=\prod_{j=1}^p(-i\alpha_jf_j)
       =\epsilon_N\Gamma_\tau\Gamma_D.
 \tag{4.5}
\] Indeed multiply the left side by \(\prod_aK_a\), which is identity
there. Moving the ambient tangent and normal words into their fixed
order gives the scalar
\(i^p(-i)^q\epsilon_p\epsilon_q i^N=(-1)^q\epsilon_p\epsilon_q=\epsilon_N\),
since \(p+q=N\) is even. The ambient volume words are
\(\Gamma_\tau\Gamma_D\). The actual total grading on the kernel is
consequently \(\epsilon_p\Gamma_{0,F}\), exactly the grading of
\(\mathscr D_V\).

There is no uncomputed kernel line. In \(S_N\widehat\otimes S_N\), the
joint equations \[
 (1\otimes\gamma_D(v)-i\Gamma_\tau\gamma_\tau(v)\otimes1)w_0=0
 \quad(v\in\mathbb R^N)
 \tag{4.6}
\] define a single unit line by the same \(N\) commuting-involution
count. It is constant, independently of \(F\) and \(\nu_i\). For
orthogonal \(v,w\), these equations give
\((\gamma_\tau(v)\gamma_\tau(w)\otimes1+1\otimes\gamma_D(v)\gamma_D(w))w_0=0\);
hence the diagonal spin connection fixes this line. The two signs of a
spin lift cancel in the tensor product. On the normal kernel,
\((\alpha_j+i f_j)w_0=0\). Creation by \((\alpha_j-i f_j)/2\) identifies
\(\Lambda F_{\mathbb{C}}\) with the whole kernel, isometrically by the
creation-annihilation relations. This identifies \(\alpha,f\) with
\(c,f\) of \(\mathscr D_V\) and intertwines the split connections. It
works under every simultaneous orthogonal tangent/normal frame change
preserving the ambient orientation, including orientation-reversing
tangent changes accompanied by normal reversals. Thus it retains the
determinant data without selecting an orientation of \(V\).

Finally use \(H+sO\), \(1\le s<\infty\), and the endpoint \(s=\infty\).
Since \((H+sO)^2=H^2+s^2O^2\), its bounded transforms converge strongly
with their adjoints to the bounded transform of \(H\) on the kernel and
the involution \(\operatorname{sgn}O\) on its complement. Their compact
defects converge in norm: outside the kernel their resolvent norm is at
most \((1+2s^2)^{-1/2}\); on the kernel it is the fixed compact
horizontal resolvent. For a smooth Clifford source section \(a\), put
\(C_a=[H,a]_{\mathrm{gr}}\). It is bounded and \(a\) preserves \(O^2\).
The integral formula
\(F_D=\pi^{-1}\int_0^\infty t^{-1/2}D(1+t+D^2)^{-1}\,dt\), together with
\([D^2,a]=DC_a+(-1)^{|a|}C_aD\), bounds its graded commutator on the
complement by a fixed multiple of \(\|C_a\|(1+2s^2)^{-1/2}\): each
resolvent term is bounded by a fixed multiple of \((1+t+2s^2)^{-1}\),
whose weighted integral is \(\pi(1+2s^2)^{-1/2}\). Thus these compact
commutators converge in norm as well. Finite spectral truncation proves
norm continuity of these compact fields at finite parameters and at
infinity. At infinity \(\operatorname{sgn}O\) graded-commutes with the
full source and is an exact odd involution, so that summand is
degenerate. The kernel summand is \(\mathscr D_V\), with grading (4.5).
This is a genuine completed interval-module homotopy, as in the
oscillator proof of the closed-index lesson, not a count of one fibre's
index. It proves the first identity in (4.2).

Multiplication by \(y_F\) and Theorem CI.6 prove the second. The full
classical closed-manifold symbol theorem of that lesson, Theorem CI.11,
says \([P]=[\![\sigma_P]\!]\otimes d_{V,-}\), retaining its \(C(V)\)
source. After forgetting that source it identifies the analytic scalar
index with \(a\otimes d_{V,-}\) for every compact relative class \(a\).
This is (2.7), proving the final assertion. No generation by finite
projective zero-operator cycles is required, and no spin\(^{c}\) or
even-dimensional hypothesis on \(V\) has been introduced. \(\square\)

\pandocbounded{\includegraphics[keepaspectratio,alt={Complement Thom, transverse correspondence and the full normal Gaussian}]{figures/KT-KK-23-embedding-index-gaussian.png}}

\emph{Figure 1.} The arrows are KK classes, with source and target as in
(2.2), (2.3), (2.6) and (2.7). The kernel calculation is (4.3)--(4.6);
its rank is \(2^p\) and its squared-oscillator gap is exactly \(2\). The
figure is a diagram of maps and operators. The construction and proofs
are Lemmas 2.1--2.4 and Theorems 3.1 and 4.1; the historical
topological-index construction is Connes, author edition, Chapter II,
Section 9. Reproducible figure and finite Clifford verification:
\href{figures/KT-KK-23-embedding-index-gaussian.py}{source}.

\subsection{5. The geometric group and its descended
assembly}\label{the-geometric-group-and-its-descended-assembly}

Write \(\tau=TV/F\), with \(q_0=\operatorname{rank}\tau\). The \(q\) in
the normal calculation was \(N-p\); \(q_0\) here is the codimension of
the foliation.

A smooth map \(f:M\to V/F\) means a genuine smooth groupoid cocycle. On
an open cover of \(M\) it consists of smooth lifts \(f_a:U_a\to V\) and
arrows \(g_{ba}:U_a\cap U_b\to G\), from \(f_a\) to \(f_b\), satisfying
the actual arrow cocycle identity. Refinement and arrow equivalence
define the same map. Holonomy derivatives glue \(f_a^*\tau\) to
\(f^*\tau\). A K-orientation is the full stable Clifford Morita module
for the virtual bundle \(f^*\tau-TM\). It can equivalently be specified
on \(TM\oplus f^*\tau\) using the canonical interleaved complex double
of \(TM\), as in the wrong-way and Thom lessons. Its line, its scalar
transition phases and its stable grading are data; equality of
determinant lines alone is insufficient.

Define \(K^{\mathrm{geom}}_{*,\tau}(BG)\) by compact closed smooth
cycles \((M,E,f)\), with \(E\) a finite complex bundle, and parity
\(\dim M-q_0\pmod2\). The three relations are direct sum/disjoint union,
compact K-oriented bordism with extending map and bundle, and even
spin\(^{c}\) vector-bundle modification. For an even spin\(^{c}\) bundle
\(W\to M\), its modification is \(\widehat M=S(W\oplus\mathbb R)\), with
coefficient \(\pi^*E\otimes b_W\). Here \(b_W\) is the actual
north/south clutch bundle: north fibre \(S_W^+\), south fibre \(S_W^-\),
glued by \(c^+(w)\) on the unit equator. Give \(\pi\) the inverse full
vertical orientation induced by the disk bundle's outward-normal-first
boundary convention. A line twist in \(S_W\) then has its actual dual in
\(\pi\)'s orientation. These are the precise conventions of
\href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-6}{K-theory
of the leaf space}, Section 5 and Lemmas 6.0ac--ad.

\textbf{Theorem 5.1 (compact smooth geometric assembly).} For this
specified geometric group there is a well-defined homomorphism \[
 \mu_{\mathrm{geom}}:K^{\mathrm{geom}}_{*,\tau}(BG)\longrightarrow K_*(A),
 \qquad [(M,E,f)]\longmapsto [E]\otimes_{C(M)}f!.
 \tag{5.1}
\] The class \(f!\) and this homomorphism retain the entire orientation
data and use reduced completions.

\textbf{Proof: the actual wrong-way class.} The complete earlier
construction is
\href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-6}{K-theory
of the leaf space}, Lemmas 6.0j--n and Theorem 6.0o. Choose an even
Euclidean embedding \(j:M\to\mathbb R^n\). The transverse holonomy germs
of the cocycle form a Hausdorff microbundle germ \(H\to M\), with
section \(M\to H\) and normal \(f^*\tau\). Its proof takes a finite
cover with closed overlap shrinkings around the compact section, chooses
representatives of the actual holonomy arrows there, and enforces all
double/triple identities on one smaller neighbourhood. The resulting
closed gluing relation and open quotient give the Hausdorff germ. It
does not identify holonomy germs while discarding their arrows.

In \(p^*\nu_j\to H\), a small neighbourhood \(N_j\) of the section has a
genuine étale map cocycle to \(V/F\times\mathbb R^n\). In transverse
coordinates its maps are \((x,y,\eta)\mapsto(y,j(x)+\eta)\); its normal
is \(\nu_j\oplus f^*\tau\). Lemma 6.0j constructs its full normal
orientation: pull back the prescribed \(TM\oplus f^*\tau\) module and
the constant Euclidean Clifford module along the orthogonal graph
quotient, then cancel the canonical complex double of \(TM\) by the two
finite Morita evaluation maps. Positive Gram square roots identify
quotient and orthogonal metrics. The same construction on an embedding
family is a full orientation family. It retains a twist by \(L\) as
\(L\), although its determinant character changes by \(L^2\).

The étale cocycle has a principal \(G\times\mathbb R^n\)-space \(P\),
with discrete countable anchor fibres. Lemma 6.0n completes its compact
chart sections to a module \(\mathcal E_g\) over
\(D_n=A\otimes C_0(\mathbb R^n)\). Its inner product is the sum over
discrete lifts of the conjugate coefficient at one endpoint times the
coefficient at the other, with the endpoint half-densities; its right
action is convolution. For a regular fibre at \(x\), the tensor map to
\(\ell^2(P_x)\) preserves that inner product. Its chart vectors are
total, so it is onto. This proves positivity and boundedness in the
reduced norm, not merely pointwise positivity of a kernel. Compact
source multiplication is compact: normalized plaque half-densities
express a coefficient supported in one small étale chart by rank-one
maps; finite chart partitions and the uniform regular Schur estimate
approximate a general compact coefficient. A countable atlas supplies a
countable dense module core. Thus \((\mathcal E_g,C_0(N_j),0)\) is an
actual KK correspondence.

Compose the normal Thom cycle, its tube extension, this correspondence
and the ordered Euclidean inverse. Retain all full Clifford normal
blocks until the specified final parity contraction. This is
\(f!\in KK_{\dim M-q_0}(C(M),A)\), in that earlier geometric
construction's fixed ordered Clifford contraction. In course
right-Clifford native notation the embedding and projection conversions
of the wrong-way lesson, Lemma 1.1, are applied together; the full
outward normal product is retained throughout. Theorem 6.0o proves its
independence of germ, cover, tube and embedding choices by actual
arrow-equivalence unitaries on \(\mathcal E_g\), essential-support
restrictions, an interval module over \(C(I,D_n)\), and the two ordered
Bott inverses. In even stabilization its middle module is the onto
tensor isometry \(\mathcal E_g\widehat\otimes C_0(\mathbb R^{2b})\).
These are additional foliation-module facts; ordinary manifold
factorization alone would not prove them.

\textbf{Proof: the three relations.} Direct sums are direct sums of the
finite coefficient modules. For an ordinary compact map \(h:L\to M\),
the actual composition and projection formulas are Theorems 6.0s and
6.0w. They use the full ordered normal quotient, the completed middle
correspondence and the ordered inverse removing the new Euclidean block
first. In particular, they apply to the following boundary inclusion and
sphere projection with their induced orientations.

For a compact bordism \((Q,E,f)\), Lemma 6.0aa constructs its smooth
double with the whole cocycle, bundle and spin\(^{c}\) structure
extended. First make the data constant in a collar by literal cocycle
pullback and full connection transport. Reflect the outward line on the
second copy. On full spin\(^{c}\) transitions this reflection is
\([s,z]\mapsto[e_1se_1,z]\); it retains the central phase, and adapted
collar transitions commute with \(e_1\). Thus the two actual principal
structures glue. The boundary \(M=\partial Q\) separates this double
\(D\). Its ordinary normal Thom unitary is
\(U(\theta(x,v))=\exp(\pi i(1+v/\sqrt{1+v^2}))\), equal to one off the
collar. It has a bounded continuous real phase \(\Phi:D\to[0,2\pi]\),
zero on the inside and \(2\pi\) on the outside. The path \(e^{it\Phi}\)
is a norm-continuous null-homotopy in \(C(D)\). Thus \(i_!(1_M)=0\) in
\(K_1(C(D))\). The projection formula and composition give
\((fi)_!([E|_M])=0\). This is Lemma 6.0y, Corollary 6.0z and Theorem
6.0ab. The incoming boundary has the full opposite grading/operator by
the reflected Clifford unitary, giving the correct integral sign.

For vector-bundle modification, the north section \(n:M\to\widehat M\)
satisfies \[
 n!\pi!=1_{C(M)},\quad n_!(1)=[b_W]-[\pi^*S_W^-],\quad
 \pi_!(1)=0,\quad\pi_!(b_W)=1.
 \tag{5.2}
\] The first identity follows by full normal/inverse-orientation
evaluation in the composition theorem. The second follows from the
entire north-disc projection \[
 \frac1{1+|v|^2}
 \begin{pmatrix}|v|^2I_-&c^+(v)\\c^+(v)^*&I_+\end{pmatrix},
\] with reference \(\operatorname{diag}(I_-,0)\), whose range is the
graph of \(c^+(v)\). Its equatorial comparison is exactly the specified
clutch. Finite bundle complements turn the varying reference into a
constant relative projection; no nontrivial spinor bundle is declared
trivial. The disk bundle is a compact bordism, so the just-proved
boundary assertion gives \(\pi_!(1)=0\). Applying the projection formula
and the first two identities gives \(\pi_!(b_W)=1\), including every
line twist. These are the full proofs of Lemma 6.0ac and Theorem 6.0ad.
In rank zero the two sphere components carry opposite stable gradings
and \(b_0=(1,0)\); their pushforward is \(a_+-a_-\), giving the same
normalization.

Consequently composition and (5.2) give \[
 (f\pi)_!(\pi^*E\,b_W)=\pi_!(\pi^*E\,b_W)\otimes f!=[E]\otimes f!.
\] All three defining relations have equal images. This proves (5.1),
exactly the descent in Theorems 6.0ae and 6.1 of that earlier lesson. No
longitudinal equality, rational character formula or assembly
isomorphism was used. \(\square\)

\subsection{6. Proper groupoids, torsion and the assembly
question}\label{proper-groupoids-torsion-and-the-assembly-question}

The smooth cycle group in Section 5 is a precise meaning of the
geometric notation \(K_{*,\tau}(BG)\). Its identification with
continuous twisted K-homology of a topological classifying space is a
further comparison theorem. In the classical torsion-free holonomy
setting, Connes' author edition, Chapter II, Section 8, Proposition 4,
describes that identification. We state that comparison and do not use
it to prove Theorem 5.1.

For arbitrary isotropy, the standard Baum--Connes domain uses
\textbf{proper} groupoid actions. Freeness must not be imposed on all
such actions: finite isotropy contributes representations. Connes,
Chapter II, Section 10, Definitions 1--3 and Theorem 7, gives the smooth
proper-cycle version and its full assembly followed by the quotient to
reduced K-theory. For locally compact second-countable Hausdorff
groupoids with Haar system, the standard equivariant version is \[
 K_*^{\mathrm{top}}(G;B)=\varinjlim_{Z\subset\underline EG,\,Z/G\text{ compact}}
 KK_*^G(C_0(Z),B),\qquad
 \mu_{r,B}(x)=[p_Z]\otimes j_r^G(x).
 \tag{6.1}
\] Here \(Z\) is a proper \(G\)-space, \(B\) a separable \(G\)-algebra,
and \(p_Z\) the cutoff projection in the transformation-groupoid crossed
product. Formula (6.1), its independence, its compatibility with the
direct limit and its comparison with the smooth geometric model are
\textbf{stated} interfaces; the exact free locator is Gómez
Aparicio--Julg--Valette, \emph{The Baum--Connes conjecture: an extended
survey}, Section 7.1, pages 79--81 of the arXiv edition. The
non-Hausdorff proper-action version requires its corresponding groupoid
hypotheses and is not obtained by applying a Hausdorff-only formulation
without change. None of (6.1)'s unproved model comparisons is a premise
of the embedding index, the special cases or the descended map (5.1).

The Baum--Connes assertion for a specified groupoid and coefficients is
that its proper-domain map \(\mu_{r,B}\) is an isomorphism. Without
coefficients take \(B=C_0(G^{(0)})\). This cannot be stated as a true
unrestricted theorem for all foliations. Higson--Lafforgue--Skandalis,
\emph{Counterexamples to the Baum--Connes conjecture}, Section 4, pages
341--344, construct a compact foliated bundle with non-Hausdorff
holonomy groupoid whose reduced assembly is neither injective nor
surjective. Their author edition has the explicit groupoid quotient, the
reduced-algebra decomposition and the K-theory diagram. This
counterexample is a \textbf{stated result}, unused in our proofs. It
disproves the blanket reduced assembly-isomorphism assertion in that
scope, while leaving (3.1) intact.

There are substantial positive classes. The Haagerup/amenable groupoid
assembly theorem and its coefficient scope are stated in the free
survey, Section 7.1, Theorem 7.2 and the paragraph following it, with
Tu's exact primary theorem locators. These are statements, not
hypotheses silently supplied for an arbitrary holonomy groupoid.
Theorems concerning groups alone, group actions with coefficients,
Hausdorff groupoids and non-Hausdorff foliations have different scopes.

\subsection{7. Examples}\label{examples}

\textbf{A submersion.} Let \(r:V\to B\) be a proper smooth submersion of
compact manifolds with connected fibres, and let \(F=\ker dr\). Plaque
holonomy is trivial and any pair in a fibre can be joined there. Thus
\(G=V\times_BV\). The completed fibre module \(E_r=L^2(V/B)\) gives
\(A=\mathcal K(E_r)\sim_M C(B)\), by the proved foliation-algebra
lesson, Propositions 16.3--16.4. The analytic index is consequently a
class of \(K^0(B)\), even when kernel dimensions jump and there is no
honest kernel bundle. The equality with the embedding-defined family
topological index is the \textbf{families index theorem, stated} as the
submersion case of (3.1). It is not deduced from merely counting the
index of each fibre. At \(B=\mathrm{pt}\), Section 4 proves that case in
full.

\textbf{Kronecker leaves.} On \(\mathbb T^2\), take the flow
\((x,y)\mapsto(x+t,y+\theta t)\), with irrational \(\theta\). A circle
transversal has return groupoid \(\mathbb T\rtimes_\theta\mathbb Z\).
Its complete-transversal Morita equivalence with \(G\) is Theorem 16.2
of the foliation-algebra lesson. Thus \(A\sim_M A_\theta\), with
\(VU=e^{2\pi i\theta}UV\) in the rotation algebra. The familiar
computation \(K_0(A_\theta)\cong\mathbb Z^2\) and the longitudinal Dirac
index descriptions are \textbf{stated} here; the precise free programme
locator is that lesson, formulas (16.36)--(16.38), using the proved
Pimsner--Voiculescu boundary conventions. A rank-one leaf Dirac is an
odd class, so its bare longitudinal symbol belongs to \(K_c^1(F^*)\),
not \(K_c^0(F^*)\). Concretely the leaf operator is
\(D_\theta=-i(\partial_x+\theta\partial_y)\), on leaf half-densities.
Its odd analytic symbol class is \(1_{\mathbb T^2}\boxtimes b_1\). The
stated free-flow comparison (7.1) identifies its index with
\(\operatorname{CT}_\theta([1_{\mathbb T^2}])\in K_1(A)\), which is
nonzero because that map is an isomorphism and the unit has rank one.
This is the explicit longitudinal Dirac statement; its locator is
Connes, author edition, Chapter II, Section 9, pages 139--140, rather
than the rotation-algebra group calculation alone. Even
coefficient/suspended symbols give the even index group. This parity
distinction is needed when comparing a displayed \(K_0\) calculation
with a one-dimensional operator.

\textbf{A free real flow.} Let \(\mathbb R\) act smoothly and freely on
compact \(V\), with generator \(X\). Every orbit with its intrinsic leaf
topology is a copy of \(\mathbb R\); the time from one endpoint to
another is unique. Leaf paths are homotopic with fixed endpoints to that
time path, so their holonomy arrows are exactly the action arrows. Flow
charts identify their smooth structures. Consequently
\(G\cong V\rtimes\mathbb R\), and \(A=C(V)\rtimes\mathbb R\), by the
groupoid/crossed-product identification in the foliation-algebra lesson,
Section 16.5. Locally free actions alone would not justify this
identification.

The positive flow coordinate trivializes \(F^*\cong V\times\mathbb R\).
Choose the positive fibre Bott suspension \(b_1\in K_c^1(\mathbb R)\) in
the same ordered right-Clifford convention as the real-action Thom map.
The free-flow index/Thom comparison is \textbf{stated}: \[
 \operatorname{Ind}_a(x\boxtimes b_1)=\operatorname{CT}_\alpha(x),
 \qquad x\in K^j(V),\qquad
 \operatorname{CT}_\alpha:K^j(V)\xrightarrow{\cong}K_{j+1}(C(V)\rtimes\mathbb R).
 \tag{7.1}
\] The exact free locator is Connes, author edition, Chapter II, Section
9, pages 139--140, the comparison of the three index/Thom maps; Connes'
author survey, Section 12, gives its historical setting. Orienting the
coordinate oppositely changes the Bott and Thom choices together.
Formula (7.1) is the statement-level comparison used in Exercise 3, not
a proof of (3.1) or a new input to Theorem 5.1.

\subsection{8. Four exercises with
solutions}\label{four-exercises-with-solutions}

\textbf{Exercise 1.} For a zero-dimensional foliation identify both
indices with identity on \(K^0(V)\).

\textbf{Solution.} Its groupoid is the unit groupoid, hence \(A=C(V)\),
and \(F^*=V\). An analytic relative class is the finite bundle
difference \([E^+]-[E^-]\); every map between the finite bundle modules
is compact, and its Fredholm index is that difference. Topologically the
normal is \(V\times\mathbb R^{2m}\), and the complete product is
\(1_{C(V)}\boxtimes\beta_{2m}\eta_{2m}=1_{C(V)}\), after the
translation/radial tube homotopy. These are the two independent
calculations of Theorem 3.1. Thus every bundle difference maps to
itself.

\textbf{Exercise 2.} Prove embedding independence of
\(\operatorname{Ind}_t\) using the full wrong-way theorem.

\textbf{Solution.} Put two embeddings in separate even blocks and use
\(i_\theta=(\cos\theta i_0,\sin\theta i_1)\). Each is an embedding. The
normal graph quotient, positive Gram maps and full ambient Clifford
modules form one smooth orientation family. Theorem 8.1 of the wrong-way
lesson and the parameter tubular construction give one normal Thom/tube
homotopy. Lemma 2.3 supplies the actual completed interval foliation
correspondence; ordinary wrong-way homotopy without that module would be
incomplete. At the endpoints the extra even normal block gives
\(\beta_{\mathrm{extra}}\), the correspondence is the old one tensored
with its identity coordinate module, and the projection gives its
ordered \(\eta_{\mathrm{extra}}\). Lemma 2.4 and Proposition 7.2 of the
wrong-way lesson cancel both factors, preserving the outward/native
\(\epsilon_r\) conversions. Exchanging two even blocks preserves the
Bott orientation. Thus the endpoint classes in \(KK(B_F,A)\) agree
before any K-class is tested, and their index homomorphisms agree.

\textbf{Exercise 3.} For a free real flow, relate the analytic index on
the trivial symbol class to Connes--Thom, at statement level.

\textbf{Solution.} The trivial bundle alone pulled back from \(V\) has
no compact support at fibre infinity. The intended symbol is its fibre
Bott Thom class \(1_V\boxtimes b_1\in K_c^1(F^*)\). The rank-one parity
is essential. Statement (7.1), with the positive flow and the same
right-Clifford orientation, gives
\(\operatorname{Ind}_a(1_V\boxtimes b_1)=\operatorname{CT}_\alpha([1_V])\in K_1(A)\).
More generally it gives the formula there for any \(x\in K^j(V)\). If
the question is written solely with \(K_c^0(F^*)\to K_0(A)\), its input
must be an odd coefficient \(x\in K^1(V)\), or the entire construction
must be suspended once. One cannot insert the even unit into that
degree-zero source. This is a check of the stated free-flow comparison
and its parity, not an independent proof of the Connes--Thom theorem.

\textbf{Exercise 4.} Explain why analytic index and assembly answer
different questions. Give an example where the geometric group differs
from \(K_c^0(F^*)\).

\textbf{Solution.} Analytic index takes a compact relative cotangent
symbol on the fixed foliated manifold to its operator class. Assembly
takes all compact K-oriented maps to its leaf space, modulo geometric
relations, to their wrong-way classes. The first domain retains symbol
data which the leaf-space cycle group can identify. For a concrete
example take \(V=\mathbb{CP}^1\) with \(F=TV\). The symbol domain is \[
 K_c^0(T^*\mathbb{CP}^1)\cong K^0(\mathbb{CP}^1)\cong\mathbb Z^2.
 \tag{8.1}
\] The first is the proved complex Thom equivalence; the second is the
integral circle/disk clutch calculation for \(S^2\) in
\href{supporting/relative-k-foundations/relative-k-foundations.html\#rk-sphere-ring}{Relative
difference bundles, radial pairs and the planar normalization}, Theorem
RK.7, also applied in the Thom lesson, Section 9, and the closed-index
lesson, Section 8, with the trivial line and the positive reduced clutch
generator. It is an integral calculation.

For the pair groupoid any cocycle is arrow-equivalent to the constant
cocycle at one chosen point: on each chart use the unique pair arrow
from that point to its lift; uniqueness makes every overlap identity
exact. Its transverse bundle is zero. The geometric group is therefore
exactly the ordinary compact spin\(^{c}\) cycle group of a point, with
the same three relations, not a new group inferred merely from a
homotopy type. That group has
\(K_0^{\mathrm{geom}}(\mathrm{pt})=\mathbb Z\) and
\(K_1^{\mathrm{geom}}(\mathrm{pt})=0\). The exact earlier proof is
\href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-6}{K-theory
of the leaf space}, Lemmas 8C.3--8C.7 and
\href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-8c}{Theorem
8C.8}, applied to the trivial graph action on a point. To see the
substantive geometric step, embed a compact cycle in \(\mathbb R^N\),
with \(N\equiv\dim M\pmod2\); its even full graph-normal Thom
coefficient is \(y\in K_c^0(\mathbb R^N)\). A compact carrier double
represents the inverse Bott coefficient \(y\eta_N\). The
embedding-transfer bordism in Lemma 8C.5 removes a disk
\(D(\nu\oplus\mathbb C\oplus\mathbb R)\) from \(D^3\times Z\); the two
ends are the even modifications of the original cycle and its carrier
cycle. The full Thom coefficient is extended only on its tube, using
finite complements to give a constant reference projection on the
complement. Its line and the inverse carrier line cancel by actual
Morita evaluation. Thus the two cycles agree by exactly the permitted
bordism and modification relations. The two Bott inverses then give the
two-sided normalization: in even degree the integer coefficient is a
signed sum of point cycles; in odd degree \(K_1(\mathbb C)=0\) gives
zero. This is the geometric second inverse, not just an analytic
numerical index.

Under pair-groupoid Morita, the point generator assembles to
\(1\in K_0(\mathbb C)\), by its finite zero-operator module. Hence in
this example the even geometric domain is \(\mathbb Z\) while the even
symbol domain is \(\mathbb Z^2\). The index is a homomorphism out of the
latter; the closed-index computation gives index \(k+1\) for the
\(\mathcal O(k)\)-twisted Dolbeault symbol. Assembly of point cycles is
already the identity integer map. Neither the equality of
analytic/topological indices nor this example identifies the two
domains. \(\square\)

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

The full longitudinal index theorem (3.1) is stated with
Connes--Skandalis Theorem 4.14 and Connes Chapter II, Section 9, Theorem
6. We prove its zero-foliation and one-leaf cases; the latter includes
every closed odd or non-spin\(^{c}\) base. The families theorem in
Section 7 is the stated general submersion case.

The classifying-space identification of the compact smooth twisted cycle
model, the general proper-equivariant domain and assembly comparison,
and the positive assembly-isomorphism theorems are stated with Connes
Chapter II, Sections 8 and 10, and Gómez Aparicio--Julg--Valette Section
7.1. The smooth geometric map (5.1) itself is proved from the actual
earlier completed correspondence and geometric-relation proofs. No
general equivariant groupoid product or proper-model identification is
used to establish it.

The HLS foliation counterexample is stated with Section 4 of its free
author edition. The rotation-algebra K-group calculation and
longitudinal Dirac descriptions are stated with the foliation-algebra
lesson, formulas (16.36)--(16.38). The Connes--Thom/free-flow comparison
(7.1) is stated with Connes Chapter II, Section 9, pages 139--140. These
statements are not premises of the embedding-independence proof, the
special-case proofs or geometric-cycle descent.

\subsection{Free mathematical reading}\label{free-mathematical-reading}

\begin{itemize}
\tightlist
\item
  Alain Connes, \emph{Noncommutative Geometry},
  \href{https://alainconnes.org/wp-content/uploads/book94bigpdf.pdf}{author
  electronic edition}, Chapter II, Sections 8--10, especially Lemma 8.2,
  Theorem 9.6 and Definitions 10.1--10.3/Theorem 10.7. Page numbers
  refer to this actual author edition.
\item
  Alain Connes, \emph{A survey of foliations and operator algebras},
  \href{https://alainconnes.org/wp-content/uploads/foliationsfine.pdf}{author
  text}, Sections 9--12, especially Theorem 4 in Section 11. The 1982
  index comparison posed in Section 10 is the theorem later proved by
  Connes--Skandalis.
\item
  Alain Connes and Georges Skandalis, \emph{The longitudinal index
  theorem for foliations},
  \href{https://ems.press/content/serial-article-files/42241}{publisher-open
  original}, Section IV, Theorems 4.11 and 4.14, Corollary 4.12 and
  Remark 4.15. The statement of the index theorem is 4.14; 4.11 is the
  preceding wrong-way independence theorem.
\item
  María P. Gómez Aparicio, Pierre Julg and Alain Valette, \emph{The
  Baum--Connes conjecture: an extended survey},
  \href{https://arxiv.org/abs/1905.10081}{arXiv author edition},
  Sections 7.1--7.2. This distinguishes proper groupoid domains,
  positive classes and counterexamples.
\item
  Nigel Higson, Vincent Lafforgue and Georges Skandalis,
  \emph{Counterexamples to the Baum--Connes conjecture},
  \href{https://vlafforg.perso.math.cnrs.fr/files/counterex.pdf}{author
  copy}, Section 4, pages 341--344. The counterexample here concerns
  non-Hausdorff holonomy groupoids and the reduced map.
\item
  Earlier full programme proofs:
  \href{../KT-CP/KT-CP-16.html}{C*-algebras of foliations and their
  Morita equivalences}, Theorem 16.2 and Propositions 16.3--16.4;
  \href{../NCG-FOLIATIONS/hilbert-modules-and-fields-on-the-leaf-space.html\#section-6}{Hilbert
  modules and fields on the leaf space}, Lemma 6.5 and Theorem 6.6;
  \href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-6}{K-theory
  of the leaf space}, Lemmas 6.0j--n, Theorems
  6.0o/6.0s/6.0w/6.0ab/6.0ad/6.0ae/6.1, and the point specialization of
  \href{../NCG-FOLIATIONS/k-theory-of-the-leaf-space.html\#section-8c}{Theorem
  8C.8}; the Bott, Thom, closed-index and ordinary wrong-way lessons
  with the locators given above.
\end{itemize}

\end{document}