Papers

Our previous work established the mapping cone Chern character on the even-degree side, using the powers of the mapping cone curvature associated with the given mapping cone connection. In this paper, we carry out the odd-degree version using a trivial mapping cone connection and an automorphism on a trivial complex vector bundle. This version is based on the related odd-degree transgression.

We study operator systems arising from Peter-Weyl and Dirac spectral truncations of $\mathrm{SU}(2)$ in the sense of Connes-Van Suijlekom. We determine operator system invariants for these systems, including the $C^*$-envelope and uniform bounds on the propagation number. For Peter-Weyl truncations, we find $\mathrm{prop} \leq 2$ for all connected compact Lie groups, generalizing known results for the torus. We also carry out an analysis of the spectral localizer by Loring and Schulz-Baldes on $\mathrm{SU}(2)$. In this analysis, we find numerical evidence that an index map for operator systems previously proposed by Van Suijlekom only captures the right information in a suitable interval of the parameter $\kappa$, instead of the limit $\kappa \downarrow 0$.

Let $f,g\in\mathbb{Z}[t]$ be products of cyclotomic polynomials such that either $g/\gcd(f,g)$ or $f/\gcd(f,g)$ (or both) is squarefree. We give explicit formulae for the invariant factors of $f(C_g)$, where $C_g$ is the companion matrix of $g$, expressing them in terms of the indices and multiplicities of the cyclotomic factors of $f$ and $g$. A more compact formula holds for the circulant case $g(t)=t^n-1$, assuming that every cyclotomic factor shared by $f$ and $g$ has multiplicity $1$ in $f$. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, $K$-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when $g$ is squarefree, we determine the least eventual period of the Smith forms of powers of $f(C_g)$.

We propose a of definition conjugate stack for a class of suitable $\mathbb{E}_3$-$\mathrm{MU}$-algebras that includes the truncated Brown-Peterson spectra $\mathrm{BP}\langle n \rangle$ for all heights $n$. Then we calculate part of $\mathrm{BP}\langle n \rangle^{\mathrm{conj}}$ as a quotient stack in terms of Witt vectors. To perform this computation we study generalized Thom spectra and their relative Topological Hochschild Homology. We exhibit the Eilenberg-Mac Lane spectrum $\mathrm{H}\mathbb{F}_2$ as a $\mathbb{E}_3$-$\mathrm{MU}$-Thom spectrum, the $p$-local Eilenberg-Mac Lane spectra $\mathrm{H}\mathbb{Z}/p^n$ (except for $\mathrm{H}\mathbb{Z}/4$) and $\mathrm{H}\mathbb{Z}_{(p)}$ as $\mathbb{E}_3$-$\mathrm{MU}_{(p)}$-Thom spectra, and the connective complex K-theory spectrum $ku$ as $\mathbb{E}_1$-$\mathrm{MU}$-Thom spectra. We accompany these “go" results with “no-go" results that show these “go" results do not admit $\mathbb{E}_{\infty}$-refinements.

We show by explicit counterexample that the Becker–Gottlieb transfer cannot be made functorial. A different counterexample using rational homotopy theory was discovered concurrently by the third author and Robin Stoll.

We prove that the Becker-Gottlieb transfer is not functorial, even on rational cohomology. More precisely, for any even $n \ge 4$, we construct a fibration $f \colon X \to S^{n+1}$ with fiber $S^2 \vee S^n$, and an oriented $S^1$-bundle $g \colon Y \to X$ such that the transfer of $f \circ g$ is non-trivial on rational cohomology. Since the transfer of an oriented $S^1$-bundle vanishes, this implies that the transfer of $f \circ g$ and the composite of the transfers of $f$ and $g$ induce different maps on rational cohomology and are thus not homotopic. The proof that the transfer of $f \circ g$ is non-trivial uses the rational models for the Becker–Gottlieb and THH transfers obtained in previous work of the authors. In the appendix, we provide a second, more involved, proof that relies on constructing explicit $C_\infty$-algebra models for odd spherical fibrations (including circle bundles), which might be of independent interest. Concurrently, another counterexample (of very different flavor) was found by Georg Lehner, Cary Malkiewich, and the first author.

For every countable discrete group, we construct a character for finite-dimensional unitary quasi-representations, with values in the inverse limit of the equivariant $K$-theory of its Rips complexes. The character is a complete invariant for quasi-representations up to stable equivalence for amenable groups and, more generally, for groups satisfying the Haagerup property and QR-quasidiagonality. For Haagerup groups satisfying quantitative QR-quasidiagonality, we also obtain a quantiative version of the theorems. These conclusions apply when the maximal group $C^*$-algebra is quasidiagonal and has the lifting property, including amenable groups, free groups, and closed orientable surface groups. For surface groups, rank and winding number give complete invariants for the classification. Finally, composing with the equivariant Chern character gives a delocalized cohomological obstruction to stable equivalence of a quasi-representation with genuine representations.

We present an $\infty$-categorical refinement of classical $KK$-theory and $E$-theory for graded, $G$-equivariant for a locally compact second countable group $G$, Real $C^*$-algebras. As a consequence, we obtain a couple of fiber sequences connecting the different $K$-theory spectra, and explicitly analyse the induced endomorphisms on $K_0$-groups. We also answer precisely some folklore questions in the area. Among other things, we show that graded $KK$- and $E$-categories are not a Dwyer-Kan localization of graded $C^*$-algebras.

Using the geometric realization spectral sequence and the Serre spectral sequence, we compute $K_1(\operatorname{Mat}_*)$, the first K-theory group of the proto-exact category of pointed matroids, answering a question of Jun, Sistko and Wright.

In this paper, we introduce and study the (involutive) representation homology of involutive algebras, following the development of representation homology of associative algebra in our earlier papers. Our main motivation is to understand the hyperoctahedral homology of involutive algebras. We show that over a field of characteristic $0$, the hyperoctahedral homology of an involutive algebra is isomorphic (as a graded commutative algebra) to its one dimensional involutive representation homology. This result enables the explicit computation of the hyperoctahedral homology of the universal enveloping algebras of involutive (DG) Lie algebras. For group algebras $k[\Gamma]$ equipped with the involution $g \mapsto g^{-1}$, we relate the involutive representation homology of $k[\Gamma]$ to the representation homology of the group $\Gamma$ with coefficients in $\mu_2$, and we show that the latter is the homology of the loop space of the mod $2$ Dold-Thom space of $B\Gamma$; more generally, we compute the representation homology of $\Gamma$ with coefficients in an arbitrary diagonalizable group scheme. As a consequence, the hyperoctahedral homology of $k[\Gamma]$ is concentrated in degree $0$ in characteristic $0$, for every $\Gamma$.

A pair of topological spaces $X$ and $Y$ is a Fourier–Mukai (FM) pair if there is an equivalence of $\infty$-categories of derived sheaves of abelian groups \[\operatorname{Shv}(X;\mathrm D(\mathbb Z))\simeq \operatorname{Shv}(Y;\mathrm D(\mathbb Z)).\] An FM pair $(X,Y)$ is non-trivial if $X$ and $Y$ are not homeomorphic. We consider the question of whether non-trivial FM pairs $(X,Y)$ exist for which $X$ and $Y$ are both "nice" spaces, which in this article means admitting a triangulation. We show that the answer is no if $X$ is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of Tate–Hochschild cohomology (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold $X$, we relate the Tate–Hochschild cohomology of $\operatorname{Shv}(X;\mathrm D(\mathbb Z))$ to the homology at infinity of $X$ (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory $\mathcal D_{\mathrm{Gor}}\subseteq\mathcal D$ called the Gorenstein locus, defined for every dualizable stable category $\mathcal D$. For a triangularizable space $X$ and a field $k$, we show that the Gorenstein locus of $\operatorname{Shv}(X;\mathrm D(k))$ is equal to the full subcategory of sheaves supported on the $k$-homology manifold locus of $X$.

Let $R$ be a Noetherian regular local ring of mixed characteristic and dimension four, and let $r$ be a member of a regular system of parameters. We prove that every finitely generated projective module over $R[1/r]$ is free. We also prove that $\mathrm{Sp}_4(R[1/r])$ acts transitively on unimodular columns of length four and that every determinant orientation of a rank-two projective module is induced by a basis. The argument reduces freeness to two vanishing results: the degree-three Zariski cohomology of the naive degree-four Milnor–Witt sheaf and the alternating Witt group of $R[1/r]$. We establish these results using integral degree-two algebraic $K$-theory, supported hermitian $K$-theory, and local motivic cohomology. A finite-rank symplectic comparison relates the image of the completion obstruction to Milnor–Witt multiplication and yields the required symplectic frames.

We revisit the relation between the Kontsevich commutative graph complex and the homology of $\GL(n,\Z)$ found by Brown–Chan–Galatius–Payne using the sharbly complex of Ash–Miller–Patzt.

We show that integers act invertibly on a derived scheme $S$ if and only if they act invertibly on the reduced motivic suspension spectrum of the affine line in $\mathrm{MS}_S$. As a consequence, over regular locally noetherian schemes and derived schemes on which some nonzero integer acts as zero, the motivic sphere spectrum is $\mathbf{A}^1$-invariant after inverting primes that are not invertible in the base scheme. In particular, over regular locally noetherian $\mathbf{Q}$-schemes, the motivic sphere spectrum is $\mathbf{A}^1$-invariant.

We prove invariance of the geometric Arveson-Douglas conjecture under Veronese embeddings of projective varieties. This provides new examples of singular varieties satisfying the conjecture. Furthermore, it reduces the conjecture to projective varieties whose defining ideals admit quadratic Gröbner bases, by keeping the same Schatten regularity thresholds. Our results lead to canonical examples of subproduct-systems whose Toeplitz algebras are not $KK$-equivalent to the complex numbers. Finally, we show that Douglas' fundamental class conjecture, asserting that the associated Toeplitz extension implements $\mathbb{T}$-equivariant duality in $KK$-theory, is invariant under Veronese embeddings as well. While the latter result bootstraps the fundamental class conjecture, it also leads to infinitely many counterexamples.

We introduce a secondary index map from relative K-homology to the K-theory of a relative structure algebra. We prove an algebraic Mayer-Vietoris sequence for the relative structure algebra, and prove compatibility of the secondary index map with the Mayer-Vietoris boundary map. As an application, we compute the secondary index of the Dirac operator on an even-dimensional spin manifold with collared metric having uniformly positive scalar curvature on the boundary. This computation leads to a refinement of the delocalized APS-index theorem.

For a proper cocompact action of a finitely generated discrete group $\Gamma$ on a spin manifold $M$, the higher localised $\widehat{A}$-genera $\widehat{A}_g(M,\omega)$ introduced earlier by the author and Mathai were shown to be an obstruction to $\Gamma$-invariant metrics of positive scalar curvature, for $\omega$ arising from the subring of $H^*(\underline{B}\Gamma,\mathbb{R})$ generated by elements of degree at most $2$, under the assumptions that the fixed-point set $M^g$ is connected and that $g$ is regular with respect to a certain real group $2$-cocycle. We show that both assumptions can be dropped.

We compute the integral Batalin-Vilkovisky (BV) algebra $\mathbb{H}_*(L\mathrm{Sp}(n);\mathbb{Z})$ of the free loop space of the symplectic group. We then show that for a connected compact Lie group $G$ whose homology over a commutative ring $R$ is exterior on odd primitive generators, $\mathbb{H}_*(LG;R)$ is the BV algebra of the free loop space of the product of odd spheres with the classical exponents of $G$, tensored with the group ring of the torsion of $\pi_1(G)$. Conversely, over an integral domain the graded algebra $\mathbb{H}_*(LG;R)$ already determines the exponents and the group ring, hence the BV algebra. If moreover $G$ is simply connected, then $\mathbb{H}_*(LG;R)$, as a BV algebra, is the Hochschild cohomology of $H^*(G;R)$ with the BV operator induced by Poincaré duality.

In their seminal work, Baum-Douglas-Taylor showed that under a certain compactness assumption, local boundary conditions for first-order elliptic operators over compact manifolds with boundary define cycles for relative K-homology. Using a systematic approach to regular boundary conditions, we generalize their result to elliptic operators of arbitrary order and to non-compact manifolds with potentially non-compact boundary. In addition, we show that under stronger regularity conditions, local boundary conditions even define cycles for absolute K-homology. In contrast to the relative classes, the resulting absolute K-homology classes are in general not independent of the choice of boundary condition.

Assuming a weak form of resolution of singularities in positive characteristic, we show several at-the-characteristic numerical invariants, including Hodge numbers, are given by homomorphisms out of the Grothendieck ring of varieties. Under the same assumption, we obtain two consequences: (i) $\widehat K$-equivalent varieties have the same Hodge numbers, answering a question of de Fernex–Mere; and (ii) universally homeomorphic varieties need not have the same class in the Grothendieck ring of varieties, answering a question of Nicaise–Sebag. In the appendix, we provide unconditional evidence for (i) by proving that birational Calabi–Yau varieties of dimension at most four have the same Hodge numbers.

A regular complex bundle on an orbispace has fibres that are positive multiples of the stabilizers' regular representations. For stabilizers of order at most two, the sharp uniform rank bound in dimension five is $16$ for paracompact orbispaces and $4$ for closed oriented smooth orbifolds. Rank $16$ occurs with dense free locus and minimum regular rank $4$ on the nonfree locus. We also prove the sharp bound $16$ for closed oriented smooth six-orbifolds. We compute the stable regular index of cyclic gerbes over finite connected CW complexes as the least common multiple of the indices of their character twists. For paracompact orbispaces, optimal integral Adams multipliers give explicit rank bounds in terms of dimension and stabilizer orders. The least positive rank of an integral exterior-power expression with regular fibre and zero reduced class for every regular rank-four bundle on $S^4\times B\mu_2$ is $32$.

We use the notion and properties of topological graphs to construct a quantum CW complex for quantum real projective spaces. This construction mirrors the CW complex construction in the classical case. Moreover, we introduce multipullback even-spheres as boundaries of noncommutative "cylinders" and show that they admit a gluing akin to the gluing of balls to the ends of a hollow closed cylinder.

We compute $K$-theoretic invariants for four three-dimensional operator systems to illustrate the general theory. We find that in some cases the invariants distinguish the operator system from its $C^*$-envelope, while in others they do not. We also explore extensions of the theory.

We compute $K$-theoretic invariants for several finite-dimensional operator systems. These include the Toeplitz operator system and its dual (the Fejér-Riesz operator system), as well as operator systems associated to tolerance relations, aka graph operator systems. A variety of techniques is applied, adapted to each of these cases, ranging from a generalization of Carathéodory's factorization result for block Toeplitz matrices, to Wiener-Hopf factorization and Schur complements. In most instances we find that the invariants coincide with their analogues for the $C^*$-envelope, except for the $K_1$-invariants of the Fejér-Riesz operator system.

For a field $F$ we study the coinvariants for the $SL_n(F)$-action on the double Steinberg module $St_n(F) \otimes St_n(F)$ and show they have a rich algebraic structure: for $n = 2$ it is the Grothendieck-Witt group of $F$, and for all $n$ they assemble to a graded nonunital $\mathbb{Z}[F^\times]$-algebra, whose rational (underived) indecomposables may be expressed in terms of the augmentation ideal of the Grothendieck-Witt group. We then explain, building on work of Galatius-Kupers-Randal-Williams, that the special linear groups $SL_n(F)$ assemble to an $E_\infty$-algebra in a suitable functor category, whose $E_2$-homology groups have a vanishing line of slope 2 and on the critical line are given by the double Steinberg coinvariants.

Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash–Miller–Patzt, uses the sharbly resolution. The other, by Brown–Chan–Galatius–Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.

This is the first in a series of papers extending Roe algebras and coarse index theory to the groupoid-equivariant setting. We introduce some techniques to develop a framework of Roe algebras and their K-theory for spaces equipped with proper actions of étale groupoids. For a fixed étale groupoid G and G-C* -algebra A, we construct a functor KC(-; G, A) from the category of locally compact Hausdorff proper G-spaces and equivariant proper continuous maps to the category of graded abelian groups, which provides a natural receptacle for an equivariant coarse index map. We study the existence of universal modules, an analog of ample modules in non-equivariant setting, and develop a decomposition technique to establish their existence. As an application, we prove that groupoid simplicial complexes satisfying suitable hypotheses admit universal modules.

Let $\overline{\mathfrak{F}}$ be the group class obtained from the class $\mathfrak{F}$ of finite groups by iterated applications of Kropholler's operation ${\scriptstyle{{\bf LH}}}$ and Talelli's operation $\Phi$. We describe the structure of the cokenels of acyclic complexes of projective modules over the group algebra of $\overline{\mathfrak{F}}$-groups with coefficients in a commutative ring. If $G$ is an $\overline{\mathfrak{F}}$-group, we show that any acyclic complex of projective (respectively, flat) $\mathbb{Q}G$-modules is contractible (respectively, pure acyclic). If $G$ is torsion-free, the same conclusions hold for acyclic complexes of projective or flat $\mathbb{Z}G$-modules. Analogous results are valid for acyclic complexes of injective modules. We present some applications regarding modules that admit complete resolutions, groups with periodic cohomology after some steps and the relation between Gorenstein and ordinary homological dimensions for modules over group algebras.

In a restricted untwisted sector, Diaconescu, Moore & Witten (2000, 2003) gave a quantum M-theoretic consistency check, at the level of partition functions, of what one may call "Hypothesis K", namely that magnetic and electric type IIA brane charges are jointly quantized in K-theory. We show that the more recently discussed "Hypothesis H" — that magnetic and electric charges in 11D are jointly quantized in coHomotopy — together with a topological smallness condition on the M-theory circle, systematically implies that IIA charges are quantized in a nonabelian deformation of twisted K-theory by the previously neglected non-linear NS1-brane charge. This is essentially a derivation of twisted K-theory from IR-completed 11D SuGra, and conversely a further consistency check on Hypothesis H from Hypothesis K. We motivate and precisely state this result, discuss its structural implications, and outline the elaborate algebro-topological proof.

We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu–Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid $\mathrm{C}^*$-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu–Renault systems are constructed, each modelling the Cuntz algebra $\mathcal{O}_2$. The $\mathrm{C}^*$-algebras obtained from orbit-breaking for such systems come with explicit embeddings into $\mathcal{O}_2$. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into $\mathcal O_2\otimes\mathcal K$. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a $^*$-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit $^*$-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their $K$-groups.