# The reduced assembly map and the torus class *Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).* The Baum–Connes assembly map compares the topology of a group with the operator algebra of its regular representation. For a countable group \(\Gamma\) and the trivial coefficient algebra, its reduced form is a homomorphism \[ \mu^r_\Gamma:RK^\Gamma_*(\mathcal E\Gamma)\longrightarrow K_*(C^*_r(\Gamma)) \] from the equivariant \(K\)-homology with \(\Gamma\)-compact supports of a universal proper \(\Gamma\)-space to the \(K\)-theory of the reduced group \(C^*\)-algebra. The Baum–Connes conjecture asserts that it is an isomorphism; it is stated in [The Baum–Connes assembly map and the conjecture](course:KT-KK/KT-KK-20). For a torsion-free group the domain is the \(K\)-homology with compact supports of a classifying space \(B\Gamma\) [Valette, Section 7.4.1]. This course proves OpenAI's theorem (2026) that the map can fail to be injective, even rationally, for a finitely generated torsion-free group [OpenAI-BCI, Theorem 1.1]. **Theorem 1.1** (OpenAI). There are a finitely generated torsion-free group \(G_{\rm inj}\), an injective homomorphism \(i:\mathbb Z^2\to G_{\rm inj}\) and a class \(c\in H^2(BG_{\rm inj};\mathbb Z)\) with \[ \begin{gathered} \bigl\langle(Bi)^*c,[T^2]\bigr\rangle=1,\\ i_*(\beta)=0\ \text{ in }K_0(C^*_r(G_{\rm inj})), \end{gathered} \] where \(\beta\) is a rank-zero Bott generator of \(K_0(C^*_r(\mathbb Z^2))\). In particular, the class \(h\) of the Dirac operator of the torus, induced along \(i\), has infinite order in \(RK_0^{G_{\rm inj}}(\mathcal EG_{\rm inj})\), while \(\mu^r_{G_{\rm inj}}(h)=0\). Here \(BG\) is the bar model of a group \(G\), \(T^2=\mathbb R^2/\mathbb Z^2\) carries the standard orientation, and the map \(Bi:T^2\to BG_{\rm inj}\) and the class \(h\) are defined in Section 3. In the description of the domain by \(K\)-homology of the classifying space, \(h\) is the class \((Bi)_*[D_{T^2}]\) of [OpenAI-BCI, Section 9.2]; the proof below works with the domain of assembly directly. - The assembly map and the Bott generator (Section 1). - The plan of the course (Section 2). - The torus class (Section 3). - Proof of the theorem (Section 4). - Maximal assembly and related results (Section 5). - Named statements (Section 6). ## 1. The assembly map and the Bott generator We use the construction of assembly in [The Baum–Connes assembly map and the conjecture](course:KT-KK/KT-KK-20), with its numbering. Let \(G\) be a countable discrete group. A *stage* is a closed \(G\)-compact subspace of the universal proper \(G\)-space of Theorem A.7 there. At a free proper \(G\)-compact space \(X\), assembly is a homomorphism \[ \mu^r_X:KK^G_*(C_0(X),\mathbb C)\longrightarrow K_*(C^*_r(G)), \] natural for proper equivariant maps, \(\mu^r(f_*x)=\mu^r_X(x)\) ((A12) in Theorem A.4), and the domain of \(\mu^r_G\) is the direct limit (U6) over the stages. A free proper \(G\)-compact space maps into a stage by a cutoff map (proof of Theorem A.7), so its classes define classes in the domain. Three further results of that lesson are used. 1. For a free proper \(G\)-compact space \(X\) with compact quotient \(Y\), there is an isomorphism \(\Theta_X\) from \(KK^G_*(C_0(X),\mathbb C)\) to \(KK_*(C(Y),\mathbb C)\) ((Q5)). It is natural: for a proper equivariant map \(f:X\to Z\) of such spaces, with quotient map \(f_0\), \(\Theta_Z(f_*x)=f_{0,*}\Theta_X(x)\). The proof of Theorem A.10 establishes this naturality by pulling a cutoff back along \(f\), an argument that uses only that \(f\) is proper and equivariant. 2. (Theorem A.8, the Mishchenko comparison.) With the Mishchenko bundle \(\mathcal L_X=X\times_GC^*_r(G)\) of \(X\), \[ \begin{aligned} &\mu^r_X(x)=[\mathcal L_X]\\ &\quad{}\otimes_{C(Y)\otimes C^*_r(G)}\\ &\quad\bigl(\Theta_X(x)\boxtimes1_{C^*_r(G)}\bigr). \end{aligned}\tag{1.1} \] 3. (Proof of Theorem A.10, (R8) and (R10).) For torsion-free \(G\), the canonical trace \(\tau\) of \(C^*_r(G)\) satisfies \[ \begin{gathered} \tau_*\bigl(\mu^r_X(x)\bigr)\\ =\bigl\langle[1_Y],\Theta_X(x)\bigr\rangle . \end{gathered}\tag{1.2} \] For \(G=\mathbb Z^2\), the plane \(\mathbb R^2\) with translations is a universal proper space and is itself a cofinal stage ((U8)), and assembly is an isomorphism (Theorem A.9). By Lemma 1.1 of [Compact Bott projections and spreading](compact-bott-projections-and-spreading.md), continuous functions \(F\) on \(\mathbb T^2=(\mathbb R/\mathbb Z)^2\), with coordinates \((\vartheta,\xi)\), act as \(F(\lambda(z),\lambda(x))\) in \(C^*_r(\langle z,x\rangle)\), with \(e^{2\pi i(c\vartheta+n\xi)}\) acting as \(\lambda(z^cx^n)\). Composed with the isomorphism \(\mathbb Z^2\to\langle z,x\rangle\) that sends the ordered basis to \((z,x)\), this identifies \(C^*_r(\mathbb Z^2)\) with \(C(\mathbb T^2)\), the canonical trace becoming the integral. In the lesson [Commutative and noncommutative tori](course:KT-OPK/commutative-and-noncommutative-tori), Corollary 3.7 with parameter \(a=0\), generators \(U=e^{2\pi i\vartheta}\), \(V=e^{2\pi i\xi}\) and derivations \(\delta_1=\partial_\vartheta\), \(\delta_2=\partial_\xi\) shows that \[ \begin{gathered} K_0(C(\mathbb T^2))\to\mathbb Z^2,\\ x\mapsto\bigl(\tau_*(x),\kappa(x)\bigr), \end{gathered}\tag{1.3} \] is an isomorphism, where on a smooth projection the Chern number is \[ \kappa(p)=\frac1{2\pi i}\int_{\mathbb T^2}\operatorname{Tr}\bigl(p\,[\partial_\vartheta p,\partial_\xi p]\bigr)\,d\vartheta\,d\xi . \] A *rank-zero Bott generator* is a class \(\beta\) with \(\tau_*(\beta)=0\) and \(\kappa(\beta)=\pm1\). By (1.3) there are exactly two, and they differ by sign. **Lemma 1.1** (the Bott class of the course). For a height \(k\), let \(\mathsf p_k\) be the scaled projection (3.1) of [Compact Bott projections and spreading](compact-bott-projections-and-spreading.md), and let \(e=\operatorname{diag}(0,1)\), written \(e_\infty\) there. The class \(\beta=[\mathsf p_k]-[e]\), which is the class \(\beta\) of (1.2) of [Carriers, trimming and constant extraction](carriers-trimming-and-constant-extraction.md) and does not depend on \(k\), is the rank-zero Bott generator with \(\kappa(\beta)=1\). *Proof.* Both \(\mathsf p_k\) and \(e\) have trace one at every point, so \(\tau_*(\beta)=0\), and \(\kappa(e)=0\) because \(e\) is constant. On the chart, \(\mathsf p_k(\vartheta,\xi)=p(t,u)\) with \((t,u)=(Z_k(\vartheta-\alpha),N_k\xi)\); outside the chart \(\mathsf p_k=e\) has zero derivatives. Then \(\partial_\vartheta\mathsf p_k=Z_k\,\partial_tp\) and \(\partial_\xi\mathsf p_k=N_k\,\partial_up\), and the substitution has Jacobian \(Z_kN_k>0\), so \[ \begin{aligned} \kappa(\mathsf p_k)&=\frac1{2\pi i}\int_{\mathbb R^2}\operatorname{Tr}\bigl(p\,[\partial_tp,\partial_up]\bigr)\,dt\,du\\ &=1 \end{aligned} \] by (2.2) of the lesson on compact Bott projections, which is computed with the orientation \(dt\wedge du\). \(\square\) ## 2. The plan of the course The group is built in three stages, and each stage serves both conclusions of the theorem. 1. **A labelled tower** (lessons [Streams, coefficient pools and universal corrections](streams-coefficient-pools-and-universal-corrections.md) to [Thetas and the pruned old graph](thetas-and-the-pruned-old-graph.md)). An infinite layered graph whose vertices are bit strings and whose edges are solutions of triangular equations; two incident labels recover a vertex block by block, so long words occur once; random permutation tables make every layer incidence expand with a uniform gap; and a first-moment count removes short cycles and short bicycles. By [The Heisenberg group and its laws](the-heisenberg-group-and-its-laws.md), short closed paths are then not laws in the Heisenberg group \(\mathsf H\). 2. **Voltages and symbols** (lessons [Compact Bott projections and spreading](compact-bott-projections-and-spreading.md) to [Polynomial avoidance, norm tests and moments](polynomial-avoidance-norm-tests-and-moments.md)). Every edge receives a voltage in \(\mathsf H\) and a matrix weight, drawn so that short closed walks lift to open walks and every weighted incidence is close, in the full regular norm, to a spread Bott symbol of the torus \(\langle z,x\rangle\). 3. **The group and the two conclusions** (lessons [The private completion and the graphical group](the-private-completion-and-the-graphical-group.md) to [Parity cancellation and the finite witness](parity-cancellation-and-the-finite-witness.md)). Imposing the closed walks of the covering graph as relators gives a torsion-free group \(\Gamma\) containing \(\mathsf H\), with an integral class detecting the torus; disjoint domains in its sheets carry the Bott symbols into the reduced group algebra, where a swindle in finitely many steps, carried out on bounded families of symbols, kills the Bott class; a finite witness gives a finitely generated subgroup. ## 3. The torus class Let \(D_{T^2}\) be the Dirac operator of the flat torus \(T^2=\mathbb R^2/\mathbb Z^2\) with periodic spinors, written \(D_2\) in Section 6.3 of [The Baum–Connes assembly map and the conjecture](course:KT-KK/KT-KK-20), and \([D_{T^2}]\in KK_0(C(T^2),\mathbb C)\) its class. It is the operator of the flat torus in [Dirac classes and the cotangent Dolbeault element](course:KT-KK/KT-KK-15), whose scalar index \(\langle[1],[D_{T^2}]\rangle\) is computed there to be zero. For a countable group \(G\) and an injective homomorphism \(i:\mathbb Z^2\to G\), let \[ X_i=G\times_{\mathbb Z^2}\mathbb R^2 , \] the quotient of \(G\times\mathbb R^2\) by \(n\cdot(g,p)=(g\,i(n)^{-1},p+n)\), with \(G\) acting on the first factor from the left. It is a disjoint union of planes indexed by \(G/i(\mathbb Z^2)\), a free proper \(G\)-compact space with \(X_i/G=T^2\). Put \[ x_i=\Theta_{X_i}^{-1}[D_{T^2}]\in KK^G_0(C_0(X_i),\mathbb C), \] and let \(h_i\) be its class in the domain of \(\mu^r_G\). For \(G=\mathbb Z^2\) and the identity, \(X=\mathbb R^2\), and we write \(h_0\) for this class. Triangulate \(\mathbb R^2\) by the unit squares, each cut by its diagonal from \((u,v)\) to \((u+1,v+1)\), with the vertices of every triangle ordered by increasing coordinates; translations preserve this. The *grid map* \(\phi_i:X_i\to EG\) is the \(G\)-equivariant simplicial map to the bar model (Section 6) sending the vertex \((g,(u,v))\) to \(g\,i(u,v)\), and each ordered triangle to the corresponding ordered simplex. Its quotient is a map \[ Bi:T^2\longrightarrow BG , \] which is the composite of the grid map of the torus into \(B\mathbb Z^2\) with the cellular map induced by \(i\). For \(i\) sending the ordered basis to \((z,x)\), Lemma 3.1 of [Cone chains, torsion-freeness and the integral detector](cone-chains-torsion-freeness-and-the-integral-detector.md) shows that it carries the fundamental class to the bar cycle \([z|x]-[x|z]\). **Lemma 3.1** (assembly of the torus). \(\mu^r_{\mathbb Z^2}(h_0)=\varepsilon\beta\) for a sign \(\varepsilon\in\{1,-1\}\). *Proof.* By (U8), assembly for \(\mathbb Z^2\) is defined on \(KK^{\mathbb Z^2}_0(C_0(\mathbb R^2),\mathbb C)\), and by Theorem A.9 and (Q5) of [The Baum–Connes assembly map and the conjecture](course:KT-KK/KT-KK-20), \(\mu^r_{\mathbb R^2}\circ\Theta_{\mathbb R^2}^{-1}\) is an isomorphism from \(KK_0(C(T^2),\mathbb C)\) onto \(K_0(C^*_r(\mathbb Z^2))\). By Lemma A.9a there, \(KK_0(C(T^2),\mathbb C)\) is free with basis \(\epsilon\boxtimes\epsilon\) and \(d\boxtimes d\), where \(\epsilon\) (written \(e\) there) is the class of evaluation at \(1\) on the circle and \(d\) the class of the circle Dirac operator, and by (T3) there \([D_{T^2}]=-d\boxtimes d\). So \(\epsilon\boxtimes\epsilon\) and \([D_{T^2}]\) form a basis. The class \(\epsilon\boxtimes\epsilon\) is the class of evaluation at a point, so \(\langle[1],\epsilon\boxtimes\epsilon\rangle=1\), while \(\langle[1],[D_{T^2}]\rangle=0\). By (1.2), in the coordinates (1.3) the images of the two basis elements are \((1,k')\) and \((0,k)\) for some integers \(k,k'\). These images form a basis of \(\mathbb Z^2\), so the determinant \(k\) is \(\pm1\), and \(\mu^r_{\mathbb Z^2}(h_0)=(0,\pm1)=\pm\beta\) by Lemma 1.1. \(\square\) The sign depends on orientation and grading conventions and is not used. Lemma 3.1 is the full-torus case of the assembly formula of Emerson and Hudson for subtori, in which the dual subtorus is a point [EH, Theorem 1.3]; here it follows from the lattice assembly isomorphism. **Lemma 3.2** (induction along a subgroup). Let \(i:\mathbb Z^2\to G\) be injective and \(j:C^*_r(\mathbb Z^2)\to C^*_r(G)\) the induced inclusion. Then \[ \mu^r_G(h_i)=j_*\bigl(\mu^r_{\mathbb Z^2}(h_0)\bigr). \] *Proof.* Write \(A=C^*_r(\mathbb Z^2)\), \(B=C^*_r(G)\), \(y=[D_{T^2}]\). The Mishchenko bundle of \(X_i\) is \[ \begin{gathered} X_i\times_GB\cong\mathbb R^2\times_{\mathbb Z^2}B\\ \cong\bigl(\mathbb R^2\times_{\mathbb Z^2}A\bigr)\otimes_jB , \end{gathered} \] where the first map restricts to the plane of the identity coset, and the second sends \(([p,a],b)\) to \([p,j(a)b]\): the balancing relations and the fibrewise inner products agree because \(j\) sends each group unitary of \(\mathbb Z^2\) to the group unitary of its image, and local trivializations give dense range. On section modules this says \([\mathcal L_{X_i}]=(\mathrm{id}\otimes j)_*[\mathcal L_{\mathbb R^2}]\) in \(KK(\mathbb C,C(T^2)\otimes B)\). The cycles \((\mathrm{id}\otimes j)^*(y\boxtimes1_B)\) and \(j_*(y\boxtimes1_A)\) are unitarily equivalent: if \(y\) is represented by \((H,\pi,F)\), both are \(H\otimes B\) with the action \(\pi\otimes j\) and the operator \(F\otimes1\), the second through \((\xi\otimes a)\otimes b\mapsto\xi\otimes j(a)b\). By (1.1) for both groups and the naturality identities of Proposition 5.1 of [Homotopy, associativity, the index pairing and KK-equivalence](course:KT-KK/KT-KK-10), \[ \begin{gathered} \mu^r_G(h_i)=(\mathrm{id}\otimes j)_*[\mathcal L_{\mathbb R^2}]\otimes(y\boxtimes1_B)\\ =[\mathcal L_{\mathbb R^2}]\otimes(\mathrm{id}\otimes j)^*(y\boxtimes1_B)\\ =[\mathcal L_{\mathbb R^2}]\otimes j_*(y\boxtimes1_A)\\ =j_*\bigl([\mathcal L_{\mathbb R^2}]\otimes(y\boxtimes1_A)\bigr)=j_*\bigl(\mu^r_{\mathbb Z^2}(h_0)\bigr). \end{gathered} \] The first equality also uses (A12) for the cutoff map that defines \(h_i\). \(\square\) **Lemma 3.3** (maps into the bar model). Let \(G\) be a countable torsion-free group. 1. Every stage \(Z\) admits a continuous \(G\)-map \(F:Z\to EG\). 2. For every injective homomorphism \(i:\mathbb Z^2\to G\), any two continuous \(G\)-maps \(X_i\to EG\) are \(G\)-homotopic. In particular, the quotient map \(T^2\to BG\) of every continuous \(G\)-map \(X_i\to EG\) is homotopic to \(Bi\). *Proof.* (1) The action on \(Z\) is free, since finite stabilizers of a torsion-free group are trivial, and \(Z\) is proper and \(G\)-compact (Lemma A.7b of [The Baum–Connes assembly map and the conjecture](course:KT-KK/KT-KK-20)). The proof of Lemma A.10a there gives a continuous \(G\)-map from \(Z\) into the join of finitely many copies of \(G\), a simplicial complex whose simplices have at most one vertex \((\gamma,k)\) for each tag \(k\). Order the vertices of every simplex by their tags; tags are \(G\)-invariant, so this order is preserved by \(G\) and by passage to faces. The map sending a simplex with vertices \((\gamma_0,k_0),\ldots,(\gamma_n,k_n)\), \(k_0<\cdots