2020 | On an Index Theorem of Chang, Weinberger and Yu | Thomas Schick, Mehran SeyedhosseiniZeitschrift: Münster Journal of Mathematics (to appear)Link zur Publikation
On an Index Theorem of Chang, Weinberger and Yu
Autoren: Thomas Schick, Mehran Seyedhosseini
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.
Münster Journal of Mathematics (to appear)
2020 | Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary | Simone Cecchini, Mehran Seyedhosseini, Vito Felice ZenobiLink zum Preprint
Relative torsion and bordism classes of positive scalar curvature metrics on manifolds with boundary
Autoren: Simone Cecchini, Mehran Seyedhosseini, Vito Felice Zenobi
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.
2020 | A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory | Mehran SeyedhosseiniLink zum Preprint
A Variant of Roe Algebras for Spaces with Cylindrical Ends with Applications in Relative Higher Index Theory
Autoren: Mehran Seyedhosseini
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.