# A twisted AH algebra with a stably trivial, non-null-homotopic unitary **Complete proof — internal mathematical review completed 5 September 2026.** This manuscript has not been externally peer reviewed or formally verified. The construction and all mathematical assertions needed for the answer are given below. ## Abstract We construct a separable, simple, unital, nuclear, stably finite approximately homogeneous \(C^*\)-algebra \(A\) with \(K_1(A)=0\) and an element of order two in \(U(A)/U_0(A)\). The obstruction is a bundle cancellation failure over products of a sphere and complex projective spaces. An isomorphism of the relevant complements would produce a mapping-torus bundle with odd top Chern number. The integral \(K\)-theory of the base forces that number to be even. Twisted point evaluations make an inductive limit simple while preserving this obstruction. This gives a negative answer to Problem LIX of Schafhauser–Tikuisis–White [STW]. The subsequent [factorial construction](stw59-factorial-torsion-simple-ah-manuscript-2026-09-05.md) realizes every prescribed finite cyclic component order. The [exact group calculation](stw59-exact-factorial-component-groups-2026-09-05.md) further proves that this original algebra has entire component group \(U(A)/U_0(A)\cong\mathbb Z/2\). ## 1. Statement and conventions For a unital \(C^*\)-algebra \(B\), let \(U(B)\) be its unitary group with the norm topology and let \(U_0(B)\) be the connected component of the identity. The algebra is \(K_1\)-injective if the canonical map \[ U(B)/U_0(B)\longrightarrow K_1(B) \] is injective. All bundle isomorphisms below cover the identity on the base. The symbol \(\mathbf1^n\) denotes the trivial complex bundle of rank \(n\); an exponent \(L^{\oplus n}\) denotes a direct sum, never a tensor power. **Theorem 1.** There are a separable, simple, unital, nuclear, stably finite AH algebra \(A\) and a unitary \(v\in U(A)\) such that \[ K_1(A)=0,\qquad v\notin U_0(A),\qquad v^2\in U_0(A). \] In particular, the class of \(v\) has order exactly two in \(U(A)/U_0(A)\), and \(A\) is not \(K_1\)-injective. Here AH means an inductive limit of finite direct sums of algebras of sections of endomorphism bundles over compact finite CW complexes. Such a building block is a full corner of a matrix algebra over its base. The argument has two parts. We first prove the finite bundle obstruction, including its invariance under arbitrary automorphisms of the ambient bundle. We then construct a simple inductive system for which the obstruction is present at every stage. ## 2. The finite bundle obstruction Fix positive integers \(d_1,\ldots,d_\ell\), and set \[ Y=\prod_{j=1}^{\ell}\mathbb{CP}^{d_j},\qquad m=\sum_{j=1}^{\ell}d_j,\qquad H=\bigoplus_{j=1}^{\ell}L_j^{\oplus d_j}. \] Here \(L_j\) is the pullback of the tautological line bundle from the \(j\)-th factor. The empty product is allowed: then \(Y\) is a point and \(H=0\). Write \(h_j=c_1(L_j)\). The integral cohomology ring is \[ H^*(Y;\mathbb Z)= \mathbb Z[h_1,\ldots,h_\ell]/(h_1^{d_1+1},\ldots,h_\ell^{d_\ell+1}), \qquad |h_j|=2. \] In particular, \[ c_m(H)=\prod_{j=1}^{\ell}h_j^{d_j}, \qquad \left|\left\langle c_m(H),[Y]\right\rangle\right|=1. \tag{2.1} \] Regard \(S^5\) as the unit sphere in \(\mathbb C^3\). Let \(F\to S^5\) be the rank-two bundle \[ F_x=\{w\in\mathbb C^3:\langle w,x\rangle=0\}. \] It satisfies \(F\oplus\mathbf1\cong\mathbf1^3\). **Lemma 2 (twisted cancellation failure).** On \(S^5\times Y\), \[ F\oplus H\not\cong\mathbf1^2\oplus H. \tag{2.2} \] The proof uses the following parity calculation. **Lemma 3 (mapping-torus parity).** Put \[ M=S^5\times Y,\qquad V=\mathbf1^3\oplus H,\qquad r=m+3. \] Let \(W\) be any complex rank-\(r\) bundle over \(S^1\times M\) whose restriction to \(\{1\}\times M\) is isomorphic to \(V\). Then \[ \left\langle c_r(W),[S^1\times M]\right\rangle\in2\mathbb Z. \tag{2.3} \] **Proof.** We use the following integral normalizations. Complex \(K\)-theory gives \[ K^1(Y)=0,\qquad K^0(Y)= \bigotimes_{j=1}^{\ell} \mathbb Z[L_j]/((L_j-1)^{d_j+1}). \tag{2.4} \] In particular \(K^0(Y)\) is free abelian and is additively generated by line monomials \(\prod_jL_j^{a_j}\). The projective-space computation follows from Bott periodicity and the even-cell filtration [Hatcher, §2.3]. The external-product Künneth isomorphism applies because these groups are free [Schochet, Künneth Theorem, pp. 443–444]. Choose integral generators \(t\in H^1(S^1;\mathbb Z)\) and \(x\in H^5(S^5;\mathbb Z)\), and put \(z=t x\). The odd \(K\)-theory generators on \(S^1\) and \(S^5\) can be chosen to have odd Chern characters \(t\) and \(x\), respectively. For \(S^5\), this follows by suspension from the Bott generator on \(S^6\), whose degree-six Chern character is an integral generator [Hatcher, Proposition 4.3]. In particular the coefficient here is \(1\), not \(2\); the corresponding top Chern class on \(S^6\) is twice a generator. Let \(p:S^1\times M\to M\) be the projection and let \[ \delta=[W]-[p^*V]\in K^0(S^1\times M). \] Its restriction to \(\{1\}\times M\) is zero. By the split restriction sequence and the preceding Künneth isomorphisms, \[ \ker\bigl(K^0(S^1\times M)\to K^0(M)\bigr) =K^1(S^1)\otimes K^1(S^5)\otimes K^0(Y). \] Consequently, for some \(\beta\in K^0(Y)\), \[ \operatorname{ch}(\delta)=z\,\operatorname{ch}(\beta). \tag{2.5} \] For clarity we compute every coefficient responsible for parity. The total Chern class of a virtual bundle is defined multiplicatively. The splitting principle and the power-series identity for \(\log(1+a)\) give \[ \log c(\delta)= \sum_{q\ge1}(-1)^{q-1}(q-1)!\operatorname{ch}_q(\delta). \tag{2.6} \] All products of terms on the right vanish, since they contain \(z^2=0\). Exponentiating therefore gives \[ c_q(\delta)=(-1)^{q-1}(q-1)!\operatorname{ch}_q(\delta). \tag{2.7} \] There are no terms for \(q<3\). First take \(\beta=\prod_jL_j^{a_j}\), so that \[ \operatorname{ch}(\beta)=\exp\Bigl(\sum_ja_jh_j\Bigr). \] For a multi-index \(b=(b_1,\ldots,b_\ell)\) with \(|b|=q-3\), the coefficient of \(z\prod_jh_j^{b_j}\) in (2.7) is \[ (-1)^{q-1}\frac{(|b|+2)!}{\prod_jb_j!}\prod_ja_j^{b_j} = (-1)^{q-1}(|b|+2)(|b|+1) \binom{|b|}{b_1,\ldots,b_\ell} \prod_ja_j^{b_j}. \tag{2.8} \] It is an even integer. An arbitrary \(\beta\) is an integral linear combination of these monomials, and (2.7) is linear in \(\operatorname{ch}(\beta)\). Every positive Chern class of \(\delta\) is therefore even. This is an integral divisibility assertion: the integral cohomology of \(S^1\times S^5\times Y\) is torsion-free, so the calculation in rational cohomology identifies the integral coefficients uniquely. Finally, \[ c(W)=c(p^*V)c(\delta). \] The bundle \(V\) is pulled back from \(Y\), which has dimension \(2m<2r\); hence \(c_r(V)=0\). Each remaining term in \(c_r(W)\) contains a positive Chern class of \(\delta\), and is even. This proves (2.3). \(\square\) **Proof of Lemma 2.** Give \(V=\mathbf1^3\oplus H\) its direct-sum Hermitian metric. It has unit sections \[ e(x,y)=(e_3,0),\qquad s(x,y)=(x,0). \] Their orthogonal complements are \[ e^\perp=\mathbf1^2\oplus H,\qquad s^\perp=F\oplus H. \] Suppose these complements were isomorphic. Polar decomposition of an isomorphism produces a unitary isomorphism between them. Extending it by \(e\mapsto s\) yields a unitary bundle automorphism \(g\) of \(V\) satisfying \(ge=s\). Form the mapping-torus bundle \[ W_g=(V\times[0,1])/\bigl((v,1)\sim(g^{-1}v,0)\bigr) \longrightarrow S^1\times M. \tag{2.9} \] This convention means precisely that a section with values \(e\) at \(t=0\) and \(ge=s\) at \(t=1\) descends to \(W_g\). Choose a smooth section \(\sigma\) of \(H\) transverse to zero. Let \(\chi:[0,1]\to[0,1]\) vanish near both endpoints and equal \(1\) near \(1/2\). The section \[ S(x,y,t)=\bigl((1-t)e_3+t x,\ \chi(t)\sigma(y)\bigr) \tag{2.10} \] has the required endpoint values. Its zeros occur exactly at \[ t=\tfrac12,\qquad x=-e_3,\qquad \sigma(y)=0. \] At \((-e_3,1/2)\), the derivative of its \(\mathbb C^3\)-component is the real-linear isomorphism \[ T_{-e_3}S^5\oplus\mathbb R\longrightarrow\mathbb C^3,\qquad (\xi,a)\longmapsto\tfrac12\xi-2a e_3. \] Thus (2.10) has isolated transverse zeros whose signed total is \[ \pm\left\langle c_m(H),[Y]\right\rangle=\pm1. \tag{2.11} \] When \(Y\) is a point the same calculation is just the single zero in \(S^5\times[0,1]\). The gluing does not affect this zero count. If the original \(g\) is only continuous, the section of the topological bundle (2.9) still has the displayed transverse local models near its zeros and is nonvanishing near the seam. The Euler number is the sum of these local indices. Equivalently, smooth the bundle, gluing, and section away from small neighborhoods of the zeros; sufficiently close smoothing on the remaining compact set creates no zeros. For a complex bundle the Euler class of its underlying oriented real bundle is the top Chern class [Hatcher, Proposition 3.13]. Hence (2.11) implies \[ \left\langle c_r(W_g),[S^1\times M]\right\rangle=\pm1. \] But \(W_g\) restricts to \(V\) on the chosen \(M\)-slice, so Lemma 3 says this integer is even. This contradiction proves (2.2). \(\square\) The use of an arbitrary \(g\) in this proof is essential. A nonzero \(c_m(H)\) alone would only exclude a particular contraction through a spare trivial line; Lemma 3 excludes every possible isomorphism of the complements. ## 3. A unitary detected by the bundle obstruction The fibration \[ U(2)\longrightarrow U(3)\longrightarrow S^5, \qquad a\longmapsto a e_3, \] is the frame fibration associated to \(F\). Bott's unstable calculation gives \[ \pi_4(U(2))\cong\mathbb Z/2,\qquad \pi_4(U(3))=0 \] [Bott, p. 315]. Its homotopy exact sequence shows that the boundary of a degree-one map on \(S^5\) is the nonzero element of \(\pi_4(U(2))\). Fix a based representative \[ u:S^4\longrightarrow U(2) \] of this element. By the usual hemisphere clutching description, the rank-two bundle clutched by \(u\) is \(F\). Moreover \(u^2\) is null-homotopic, and the block inclusion \(u\oplus1:S^4\to U(3)\) is null-homotopic. Put \[ X=S^4\times Y,\qquad E=\mathbf1^2\oplus H,\qquad B=\Gamma(X,\operatorname{End}(E)), \] and define \(w=u\oplus1_H\in U(B)\). **Corollary 4.** The unitary \(w\) is not null-homotopic in \(U(B)\). Its class in \(U(B)/U_0(B)\) has order exactly two, and its stable \(K_1\)-class is zero. **Proof.** Glue the pullbacks of \(E\) on \(D^5_+\times Y\) and \(D^5_-\times Y\) along \(S^4\times Y\), using \(w\). The resulting bundle on \(S^5\times Y\) is \(F\oplus H\). If \(w\) were joined to the identity by a path of unitary sections, that path would extend \(w\) over \(D^5\times Y\): collapse the identity end of the radial parameter to the center. Changing a hemisphere trivialization by this extension would identify the clutched bundle with the one obtained by identity gluing, namely \(\mathbf1^2\oplus H\). This contradicts Lemma 2. The null-homotopy of \(u^2\) supplies one for \(w^2\). Finally, \(w\oplus1_E\) acts as \(u\) on the first trivial rank-two summand of \(E\oplus E\). One trivial line from the second copy of \(E\) lets us apply the null-homotopy of \(u\oplus1\), leaving all other summands fixed. Thus the stabilized unitary is null-homotopic. \(\square\) In particular, the first nontrivial twisted point-evaluation test is decided by taking \(Y=\mathbb{CP}^2\) and \(H=L^{\oplus2}\). The unitary \(u\oplus1_{L^{\oplus2}}\) survives. This conclusion does not require computing a relative gauge-group boundary subgroup. ## 4. The inductive system Index the stages by \(i\ge0\), and set \[ r_i=2^{i+1},\qquad Y_i=\prod_{j=0}^{i-1}\mathbb{CP}^{r_j},\qquad X_i=S^4\times Y_i. \] Let \(L_{j+1}\) denote the tautological line from the factor \(\mathbb{CP}^{r_j}\), pulled back as necessary, and put \[ H_i=\bigoplus_{j=0}^{i-1}L_{j+1}^{\oplus r_j}, \qquad E_i=\mathbf1^2\oplus H_i, \qquad A_i=\Gamma(X_i,\operatorname{End}(E_i)). \tag{4.1} \] For \(i=0\), these definitions mean \(Y_0=\{\mathrm{pt}\}\), \(H_0=0\), and \(A_0=M_2(C(S^4))\). Notice that \[ \operatorname{rank}(H_i)=\sum_{j7/8\). Its image is full in a later stage. Thus the closed ideal it generates there contains the unit, which lies in \(J\). Hence \(J=A\). Each \(A_i\) is a unital separable nuclear homogeneous algebra over a compact finite CW complex. These properties pass to the indicated countable inductive limit. Infinite-dimensionality follows from the injective inclusion of \(A_0=M_2(C(S^4))\). Each \(X_i\) has a finite CW decomposition with only even-dimensional cells. Its cellular \(K\)-theory exact sequences, with Bott periodicity, give \(K^1(X_i)=0\). The bundle \(E_i\) has positive rank everywhere, so its section module implements Morita equivalence between \(A_i\) and \(C(X_i)\). Therefore \(K_1(A_i)=0\). Continuity of \(C^*\)-algebra \(K\)-theory gives \[ K_1(A)=\varinjlim K_1(A_i)=0. \] For stable finiteness, each \(A_i\) has tracial states, for example the normalized trace at any fiber. Its tracial state space \(T(A_i)\) is nonempty and weak-* compact. The restriction maps \(\varphi_i^*:T(A_{i+1})\to T(A_i)\) define an inverse system with nonempty inverse limit. Indeed every finite collection of compatibility conditions is satisfied by choosing a trace at its latest stage and restricting backwards; compactness of \(\prod_iT(A_i)\) gives a compatible family. No surjectivity of restriction maps is required. This family gives a tracial state \(\tau\) on \(A\). The null ideal \[ \{a\in A:\tau(a^*a)=0\} \] is a closed two-sided ideal, and is proper because \(\tau(1)=1\). Simplicity makes it zero, so \(\tau\) is faithful. Its matrix amplifications are faithful traces; therefore no matrix algebra over \(A\) has an infinite projection. This is stable finiteness. \(\square\) ## 6. Persistence of the nontrivial unitary Let \(v_i=\varphi_{0,i}(u)\in U(A_i)\), where \(v_0=u\). Define the canonical comparison unitary \[ w_i=u\oplus1_{H_i}\in U(A_i). \] **Lemma 6.** For every \(i\), \(v_i\) is homotopic to \(w_i\), and neither is in \(U_0(A_i)\). **Proof.** The first assertion holds at stage zero. Suppose it holds at stage \(i\). Applying \(\varphi_i\) to the homotopy shows that \(v_{i+1}\) is homotopic to \[ \pi_i^*w_i\oplus\bigl(w_i(x_i)\otimes1_{L_{i+1}}\bigr). \] The matrix \(w_i(x_i)\in U(E_i(x_i))\cong U(r_i)\) is joined to the identity by a path in this finite-dimensional unitary group. Tensoring that path with \(1_{L_{i+1}}\) contracts the second block and gives \(w_{i+1}\). Apply Corollary 4 with \(d_j=r_{j-1}\) for \(1\le j\le i\). Its \(Y,H,E\) are exactly \(Y_i,H_i,E_i\), so \(w_i\notin U_0(A_i)\). \(\square\) We include the norm-topological argument needed at the limit, since finite-stage nontriviality by itself must not be silently substituted for persistence. **Lemma 7 (finite-stage detection of a null-homotopy).** Let \(B=\overline{\bigcup_iB_i}\) be a unital increasing union of \(C^*\)-algebras. If \(a\in U(B_k)\) lies in \(U_0(B)\), then it lies in \(U_0(B_j)\) for some \(j\ge k\). **Proof.** Take a continuous unitary path \(\gamma:[0,1]\to U(B)\) from \(a\) to \(1\). Choose a partition \[ 0=t_0