Callias-type operators in C∗-algebras and positive scalar curvature on noncompact manifolds

2018 ◽  
Vol 12 (04) ◽  
pp. 897-939 ◽  
Author(s):  
Simone Cecchini
Keyword(s):  
Scalar Curvature ◽  
Riemannian Metrics ◽  
Index Theorem ◽  
Complete Riemannian Manifold ◽  
Positive Scalar Curvature ◽  
Type Operator ◽  
Spin Manifold ◽  
Compact Set ◽  
Positive Scalar ◽  
C Algebras

A Dirac-type operator on a complete Riemannian manifold is of Callias-type if its square is a Schrödinger-type operator with a potential uniformly positive outside of a compact set. We develop the theory of Callias-type operators twisted with Hilbert [Formula: see text]-module bundles and prove an index theorem for such operators. As an application, we derive an obstruction to the existence of complete Riemannian metrics of positive scalar curvature on noncompact spin manifolds in terms of closed submanifolds of codimension one. In particular, when [Formula: see text] is a closed spin manifold, we show that if the cylinder [Formula: see text] carries a complete metric of positive scalar curvature, then the (complex) Rosenberg index on [Formula: see text] must vanish.

scholarly journals On Moduli Spaces of Positive Scalar Curvature Metrics on Highly Connected Manifolds

2020 ◽  
Author(s):  
Michael Wiemeler
Keyword(s):  
Fundamental Group ◽  
Scalar Curvature ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Positive Scalar Curvature ◽  
Spin Manifold ◽  
Homotopy Groups ◽  
Positive Scalar ◽  
Simply Connected ◽  
Highly Connected Manifolds

Abstract Let $M$ be a simply connected spin manifold of dimension at least six, which admits a metric of positive scalar curvature. We show that the observer moduli space of positive scalar curvature metrics on $M$ has non-trivial higher homotopy groups. Moreover, denote by $\mathcal{M}_0^+(M)$ the moduli space of positive scalar curvature metrics on $M$ associated to the group of orientation-preserving diffeomorphisms of $M$. We show that if $M$ belongs to a certain class of manifolds that includes $(2n-2)$-connected $(4n-2)$-dimensional manifolds, then the fundamental group of $\mathcal{M}_0^+(M)$ is non-trivial.

scholarly journals Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

2020 ◽  
Vol 63 (4) ◽  
pp. 901-908
Author(s):  
Philipp Reiser
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Riemannian Metrics ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Positive Scalar ◽  
Space Of Riemannian Metrics ◽  
Simply Connected

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

scholarly journals An index theorem for end-periodic operators

2015 ◽  
Vol 152 (2) ◽  
pp. 399-444 ◽  
Author(s):  
Tomasz Mrowka ◽  
Daniel Ruberman ◽  
Nikolai Saveliev
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Differential Operators ◽  
Riemannian Metrics ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Fredholm Theory ◽  
Eta Invariant ◽  
First Order ◽  
Periodic Operators

We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

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