We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion... more We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.
Proceedings of the American Mathematical Society, 2019
We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have unifor... more We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f : X n → S k × T n − k f\colon X^n\rightarrow S^k\times T^{n-k} , with k = 1 , 2 , 3 k=1,2,3 . When X X is a closed oriented manifold endowed with a metric g g of positive scalar curvature and the map f f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g g and the contracting factor of the map f f .
Let $\Gamma$ be a finitely generated discrete group and let $\widetilde{M}$ be a Galois $\Gamma$-... more Let $\Gamma$ be a finitely generated discrete group and let $\widetilde{M}$ be a Galois $\Gamma$-covering of a smooth compact manifold $M$. Let $u:X\to B\Gamma$ be the associated classifying map. Finally, let $\mathrm{S}_*^\Gamma (\widetilde{M})$ be the analytic structure group, a K-theory group appearing in the Higson-Roe exact sequence $\cdots\to \mathrm{S}_*^\Gamma(\widetilde{M})\to K_*(M)\to K_*(C^*\Gamma)\to\cdots$. Under suitable assumptions on $\Gamma$ we construct two pairings, first between $\mathrm{S}^\Gamma_*(\widetilde{M})$ and the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and secondly between $\mathrm{S}^\Gamma_*(\widetilde M)$ and the relative cohomology $H^*(M\to B\Gamma)$. Both are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class $\rho(\widetilde{D})\in \mathrm{S}_*^\Gamma (\widetilde{M})$ of an invertible $\Gamma$-equivariant Dirac...
We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion... more We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.
We present a decomposition of rational twisted G-equivariant Ktheory, G a finite group, into cycl... more We present a decomposition of rational twisted G-equivariant Ktheory, G a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal [AS89] as well as the decomposition by Adem and Ruan for twists coming from group cocycles [AR03].
Given a manifold with corners X, we associates to it the corner structure simplicial complex ΣX .... more Given a manifold with corners X, we associates to it the corner structure simplicial complex ΣX . Its reduced K-homology is isomorphic to the K-theory of the C ∗-algebra Kb(X) of b-compact operators on X. Moreover, the homology of ΣX is isomorphic to the conormal homology of X. In this note, we constract for an arbitrary abstract finite simplicial complex Σ a manifold with corners X such that ΣX ∼= Σ. As a consequence, the homology and K-homology which occur for finite simplicial complexes also occur as conormal homology of manifolds with corners and as K-theory of their b-compact operators. In particular, these groups can contain torsion.
The recent article “On Gromov’s dihedral extremality and rigidity conjectures” by Jinmin Wang, Zh... more The recent article “On Gromov’s dihedral extremality and rigidity conjectures” by Jinmin Wang, Zhizhang Xie and Guoliang Yu makes a number of claims for self-adjoint extensions of Dirac type operators on manifolds with corners under local boundary conditions. We construct a counterexample to an index computation in that paper which affects the proof of its main result stating a generalisation of Gromov’s dihedral extremality conjecture. Consider the manifold with corners A := [−2, 2] \ (−1, 1) ⊂ R. together with the Euler characteristic operator d + d : Ω(A) → Ω(A) on smooth differential forms with grading by even/odd differential forms. Let Di (for initial operator) be the restriction of d + d to forms supported in the complement of the vertices of A and subject to absolute boundary conditions. Remark 1. Recall that (by definition) a smooth differential form ω ∈ Ω(A) supported away from the vertices satisfies absolute boundary conditions if ι(∗ω) = 0 where ∗ω is the Hodge star of ω...
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter grou... more We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions t... more In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and d...
We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion... more We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.
Proceedings of the American Mathematical Society, 2019
We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have unifor... more We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f : X n → S k × T n − k f\colon X^n\rightarrow S^k\times T^{n-k} , with k = 1 , 2 , 3 k=1,2,3 . When X X is a closed oriented manifold endowed with a metric g g of positive scalar curvature and the map f f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g g and the contracting factor of the map f f .
Let $\Gamma$ be a finitely generated discrete group and let $\widetilde{M}$ be a Galois $\Gamma$-... more Let $\Gamma$ be a finitely generated discrete group and let $\widetilde{M}$ be a Galois $\Gamma$-covering of a smooth compact manifold $M$. Let $u:X\to B\Gamma$ be the associated classifying map. Finally, let $\mathrm{S}_*^\Gamma (\widetilde{M})$ be the analytic structure group, a K-theory group appearing in the Higson-Roe exact sequence $\cdots\to \mathrm{S}_*^\Gamma(\widetilde{M})\to K_*(M)\to K_*(C^*\Gamma)\to\cdots$. Under suitable assumptions on $\Gamma$ we construct two pairings, first between $\mathrm{S}^\Gamma_*(\widetilde{M})$ and the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and secondly between $\mathrm{S}^\Gamma_*(\widetilde M)$ and the relative cohomology $H^*(M\to B\Gamma)$. Both are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class $\rho(\widetilde{D})\in \mathrm{S}_*^\Gamma (\widetilde{M})$ of an invertible $\Gamma$-equivariant Dirac...
We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion... more We construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion and define the associated push-forward which satisfies functoriality, compatibility with pull-back diagrams, and projection and bordism formulas. We construct a multiplicative lift of the Chern character from smooth K-theory to smooth rational cohomology and verify that the cohomological version of the Atiyah-Singer index theorem for families lifts to smooth cohomology.
We present a decomposition of rational twisted G-equivariant Ktheory, G a finite group, into cycl... more We present a decomposition of rational twisted G-equivariant Ktheory, G a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal [AS89] as well as the decomposition by Adem and Ruan for twists coming from group cocycles [AR03].
Given a manifold with corners X, we associates to it the corner structure simplicial complex ΣX .... more Given a manifold with corners X, we associates to it the corner structure simplicial complex ΣX . Its reduced K-homology is isomorphic to the K-theory of the C ∗-algebra Kb(X) of b-compact operators on X. Moreover, the homology of ΣX is isomorphic to the conormal homology of X. In this note, we constract for an arbitrary abstract finite simplicial complex Σ a manifold with corners X such that ΣX ∼= Σ. As a consequence, the homology and K-homology which occur for finite simplicial complexes also occur as conormal homology of manifolds with corners and as K-theory of their b-compact operators. In particular, these groups can contain torsion.
The recent article “On Gromov’s dihedral extremality and rigidity conjectures” by Jinmin Wang, Zh... more The recent article “On Gromov’s dihedral extremality and rigidity conjectures” by Jinmin Wang, Zhizhang Xie and Guoliang Yu makes a number of claims for self-adjoint extensions of Dirac type operators on manifolds with corners under local boundary conditions. We construct a counterexample to an index computation in that paper which affects the proof of its main result stating a generalisation of Gromov’s dihedral extremality conjecture. Consider the manifold with corners A := [−2, 2] \ (−1, 1) ⊂ R. together with the Euler characteristic operator d + d : Ω(A) → Ω(A) on smooth differential forms with grading by even/odd differential forms. Let Di (for initial operator) be the restriction of d + d to forms supported in the complement of the vertices of A and subject to absolute boundary conditions. Remark 1. Recall that (by definition) a smooth differential form ω ∈ Ω(A) supported away from the vertices satisfies absolute boundary conditions if ι(∗ω) = 0 where ∗ω is the Hodge star of ω...
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter grou... more We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions t... more In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and d...
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