Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On positive scalar curvature bordism

Paolo Piazza, Thomas Schick, Vito Felice Zenobi

Abstract

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.

On positive scalar curvature bordism

Abstract

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.

Paper Structure

This paper contains 5 sections, 5 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Secondary higher index
  3. Case $n\equiv 0\pmod 4$
  4. Case $n\equiv 2\pmod 4$
  5. Positive scalar curvature bordism

Key Result

Theorem 1.1

Let $n=4$ or $6$ and $\{1\}\ne G$ be a fundamental group of a $3$-dimensional spherical space form if $n=4$, or more generally a finite group with at least one element not conjugate to its inverse if $n=6$. Set $\Gamma:=G\times\mathbb{Z}$. Then $\mathop{\mathrm{Pos}}\nolimits_n^{\mathop{\mathrm{spin

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 2 more