T-structure and the Yamabe invariant

TL;DR

This paper explores the Yamabe invariant of T-structured manifolds, extending known results about product manifolds involving tori and spin manifolds with nonzero -genus to more complex T-structured bundles.

Contribution

It generalizes the understanding of the Yamabe invariant to a broader class of T-structured manifolds, including certain bundles with specific transition functions.

Findings

T-structured manifolds have zero Yamabe invariant under certain conditions.

Extension of known results from product manifolds to T-bundles.

Identification of conditions involving transition functions for the Yamabe invariant.

Abstract

The Yamabe invariant is an invariant of a closed smooth manifold, which contains information about possible scalar curvature on it. It is well-known that a product manifold T^m\times B where T^m$ is the m-dimensional torus, and B is a closed spin manifold of nonzero \hat{A}-genus has zero Yamabe invariant. We generalize it to various T-structured manifolds, for example T^m-bundles over such B whose transition functions take values in Sp(m,Z) (or Sp(m-1,Z)\oplus \pm 1 for odd m).