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Schick Thomas - One of the best experts on this subject based on the ideXlab platform.
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A note on invertibility of the Dirac operator twisted with Hilbert-A-module coefficients
2021Co-Authors: Schick ThomasAbstract:Given a closed connected spin manifold M with non-negative and somewhere Positive Scalar curvature, we show that the Dirac operator twisted with any flat Hilbert module bundle is invertible.Comment: 3 pages. v2: inaccuracies correcte
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Mapping analytic surgery to homology, higher rho numbers and metrics of Positive Scalar curvature
2021Co-Authors: Piazza Paolo, Schick Thomas, Zenobi, Vito FeliceAbstract:Let $\Gamma$ be a f.g. discrete group and let $\tilde M$ be a Galois $\Gamma$-covering of a smooth closed manifold $M$. Let $S_*^\Gamma(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_*^\Gamma(\tilde M)\to K_*(M)\to K_*(C_r^*\Gamma)\to$. We prove that for an arbitrary discrete group $\Gamma$ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology $\to H_{[*-1]}(\mathcal{A}\Gamma)\to H^{del}_{[*-1]}(\mathcal{A}\Gamma)\to H^{e}_{[*]}(\mathcal{A}\Gamma)\to$, with $\mathcal{A}\Gamma$ a dense homomorphically closed subalgebra of $C^*_r\Gamma$. Here, $ H_{*}^{del}(\mathcal{A}\Gamma)$ is the delocalized homology and $H_{*}^{e}(\mathcal{A}\Gamma)$ is the homology localized at the identity element. Then, under additional assumptions on $\Gamma$, we prove the existence of a pairing between $HC^*_{del}(\mathbb{C}\Gamma)$, the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and $H^{del}_{*-1}(\mathcal{A}\Gamma)$. This, in particular, gives a pairing between $S^\Gamma_*(\tilde M)$ and $HC^{*-1}_{del}(\mathbb{C}\Gamma)$. We also prove the existence of a pairing between $S^\Gamma_*(\tilde M)$ and the relative cohomology $H^{[*-1]}(M\to B\Gamma)$. Both these parings 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 $\rho(\tilde D)\in S_*^\Gamma(\tilde M)$ of an invertible $\Gamma$-equivariant Dirac type operator on $\tilde M$. Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of $M$. Then, we establish new results on the moduli space of metrics of Positive Scalar curvature when $M$ is spin.Comment: 103 pages. Changes from the first version: the title has been modified; imprecisions have been corrected; more details are given; several new sections with many geometric applications have been adde
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On an Index Theorem of Chang, Weinberger and Yu
2020Co-Authors: Schick Thomas, Seyedhosseini MehranAbstract: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 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 direct proof of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental groups of the manifold and its boundary into account requires working with maximal C* completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions.Comment: v2: added some clarifications on the role of the new functionral completion of the Roe algebra and the relation to the originial CWY paper. To appear in M\"unster Journal of Mathematics. 33 page
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On Positive Scalar curvature bordism
2020Co-Authors: Piazza Paolo, Schick Thomas, Zenobi, Vito FeliceAbstract:Using standard results from higher (secondary) index theory, we prove that the Positive Scalar curvature bordism groups of a cartesian product GxZ are infinite in dimension 4n if n>0 G a group with non-trivial torsion. We construct representatives of each of these classes which are connected and with fundamental group GxZ. We get the same result in dimension 4n+2 (n>0) if G is finite and contains an element which is not conjugate to its inverse. This generalizes the main result of Kazaras, Ruberman, Saveliev, "On Positive Scalar curvature cobordism and the conformal Laplacian on end-periodic manifolds" to arbitrary even dimensions and arbitrary groups with torsion.Comment: 7 pages. v2 corrected typos, added references and more details of some proofs and constructions, following suggestions of referees. To appear in Communications in Analysis and Geometr
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The Gromov-Lawson codimension 2 obstruction to Positive Scalar curvature and the C*-index
2020Co-Authors: Kubota Yosuke, Schick ThomasAbstract:Gromov and Lawson developed a codimension 2 index obstruction to Positive Scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.Comment: 12 pages. v2 final version to appear in G&T. Small corrections and minor changes of presentatio
Xie Zhizhang - One of the best experts on this subject based on the ideXlab platform.
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A quantitative relative index theorem and Gromov's conjectures on Positive Scalar curvature
2021Co-Authors: Xie ZhizhangAbstract:In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on Positive Scalar curvature. More precisely, we prove a $\lambda$-Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This $\lambda$-Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal $\lambda$-Lipschitz rigidity theorem for Positive Scalar curvature metrics on hemispheres. These give Positive answers to the corresponding open questions raised by Gromov. As another application, we prove Gromov's $\square^{n-m}$ inequality on the bound of distances between opposite faces of spin manifolds with cube-like boundaries with a suboptimal constant. As immediate consequences, this implies Gromov's cube inequality on the bound of widths of Riemannian cubes and Gromov's conjecture on the bound of widths of Riemannian bands with suboptimal constants. Further geometric applications will be discussed in a forthcoming paper.Comment: 57 pages. From v3 to v4, some substantial changes have been made, and some mistakes have been correcte
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A relative index theorem for incomplete manifolds and Gromov's conjectures on Positive Scalar curvature
2021Co-Authors: Xie ZhizhangAbstract:In this paper, we prove a relative index theorem for incomplete manifolds (e.g. the interior of a compact manifold with corners, the regular part of a compact singular manifold, or their Galois covering spaces). We apply this relative index theorem to prove several conjectures of Gromov on Positive Scalar curvature. In particular, we prove Gromov's $\square^{n-m}$ conjecture on the bound of distances between opposite faces of spin manifolds with cube-like boundaries. As immediate consequences, this implies Gromov's conjecture on the bound of widths of Riemannian cubes and Gromov's conjecture on the bound of widths of Riemannian bands. Other geometric applications of our relative index theorem include the following: a rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed (the class of subsets that are allowed is rather general, which in particular includes finite subsets); and a Positive solution to the long neck problem for distance-contracting maps to spheres. These give Positive answers to the corresponding open questions raised by Gromov. Further geometric applications will be discussed in a forthcoming paper.Comment: 84 pages. A technical problem in the first version has been corrected. More precisely, the distance-contracting condition in Theorems E, F, G, H and I (of the second version) was mistakenly stated to be area-decreasing in the first version. The proofs of these theorems have been revised accordingly in the second versio
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A proof of Gromov's cube inequality on Scalar curvature
2021Co-Authors: Wang Jinmin, Xie Zhizhang, Yu GuoliangAbstract:Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a Positive Scalar curvature metric in dimension $\leq 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube inequality in all dimensions with the optimal constant via Dirac operator method. In fact, our proof yields a strengthened version of Gromov's cube inequality, which does not seem to be accessible by minimal surface method.Comment: 19 pages. v2 to v3: minor change
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Quantitative K-theory, Positive Scalar curvature, and band width
2020Co-Authors: Guo Hao, Xie Zhizhang, Yu GuoliangAbstract:We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of Positive Scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to Positive Scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly Positive Scalar curvature, the propagation at which the index of the Dirac operator vanishes is related inversely to the curvature lower bound. Second, we give an approach, using related techniques, to Gromov's band width conjecture, which has been the subject of recent work by Zeidler and Cecchini from a different point of view.Comment: 22 page
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On the range of the relative higher index and the higher rho-invariant for Positive Scalar curvature
2019Co-Authors: Xie Zhizhang, Yu Guoliang, Zeidler RudolfAbstract:Let $M$ be a closed spin manifold which supports a Positive Scalar curvature metric. The set of concordance classes of Positive Scalar curvature metrics on $M$ forms an abelian group $P(M)$ after fixing a Positive Scalar curvature metric. The group $P(M)$ measures the size of the space of Positive Scalar curvature metrics on $M$. Weinberger and Yu gave a lower bound of the rank of $P(M)$ in terms of the number of torsion elements of $\pi_1(M)$. In this paper, we give a sharper lower bound of the rank of $P(M)$ by studying the image of the relative higher index map from $P(M)$ to the real K-theory of the group $\mathrm{C}^\ast$-algebra $\mathrm{C}^\ast_{\mathrm{r}}(\pi_1(M))$. We show that it rationally contains the image of the Baum-Connes assembly map up to a certain homological degree depending on the dimension of $M$. At the same time we obtain lower bounds for the Positive Scalar curvature bordism group by applying the higher rho-invariant.Comment: 22 pages, 1 figure; v2: added details to Section 3.2 and a new Section
Mark Walsh - One of the best experts on this subject based on the ideXlab platform.
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homotopy invariance of the space of metrics with Positive Scalar curvature on manifolds with singularities
Symmetry Integrability and Geometry-methods and Applications, 2021Co-Authors: Boris Botvinnik, Mark WalshAbstract:In this paper we study manifolds, $X_{\Sigma}$, with fibred singularities, more specifically, a relevant space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ of Riemannian metrics with Positive Scalar curvature. Our main goal is to prove that the space ${\mathcal R}^{\rm psc}(X_{\Sigma})$ is homotopy invariant under certain surgeries on $X_{\Sigma}$.
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cobordism invariance of the homotopy type of the space of Positive Scalar curvature metrics
Proceedings of the American Mathematical Society, 2013Co-Authors: Mark WalshAbstract:This is the publisher’s final pdf. The published article is copyrighted by the American Mathematical Society and can be found at: http://www.ams.org/publications/journals/journalsframework/proc.
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cobordism invariance of the homotopy type of the space of Positive Scalar curvature metrics
arXiv: Differential Geometry, 2011Co-Authors: Mark WalshAbstract:We show that the homotopy type of the space of metrics of Positive Scalar curvature on a smooth manifold remains unchanged, after application of surgery in codimension at least three to the underlying manifold. This result is originally due to V. Chernysh, but remains unpublished.
Gang Tian - One of the best experts on this subject based on the ideXlab platform.
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kahler einstein metrics with Positive Scalar curvature
Inventiones Mathematicae, 1997Co-Authors: Gang TianAbstract:In this paper, we prove that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kahler manifold admits Kahler-Einstein metrics if it has Positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kahler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable Kahler manifolds, provided that the partial C 0-estimate posed in [T6] is true.
Seyedhosseini Mehran - One of the best experts on this subject based on the ideXlab platform.
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Relative torsion and bordism classes of Positive Scalar curvature metrics on manifolds with boundary
2020Co-Authors: Cecchini Simone, Seyedhosseini Mehran, Zenobi, Vito FeliceAbstract:In this paper, we define a relative $L^2$-$\rho$-invariant for Dirac operators on odd-dimensional spin manifolds with boundary and show that they are invariants of the bordism classes of Positive Scalar curvature metrics which are collared near the boundary. As an application, we show that if a $4k+3$-dimensional spin manifold with boundary admits such a metric and if, roughly speaking, there exists a torsion element in the difference of the fundamental groups of the manifold and its boundary, then there are infinitely many bordism classes of such psc metrics on the given manifold. This result in turn implies that the moduli-space of psc metrics on such manifolds has infinitely many path components. We also indicate how to define delocalised $\eta$-invariants for odd-dimensional spin manifolds with boundary, which could then be used to obtain similar results for $4k+1$-dimensional manifolds.Comment: 18 page
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On an Index Theorem of Chang, Weinberger and Yu
2020Co-Authors: Schick Thomas, Seyedhosseini MehranAbstract: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 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 direct proof of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental groups of the manifold and its boundary into account requires working with maximal C* completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions.Comment: v2: added some clarifications on the role of the new functionral completion of the Roe algebra and the relation to the originial CWY paper. To appear in M\"unster Journal of Mathematics. 33 page
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A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory
2020Co-Authors: Seyedhosseini MehranAbstract:In this paper we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of Positive Scalar curvature on such manifolds which are collared on the cylindrical end. We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for Positive Scalar curvature metrics on manifolds with boundary. This paves the way for classification of these metrics. Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of Positive Scalar curvature on the boundary.Comment: v2: references updated, introduction slightly modified, 26 page