Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures

TL;DR

This paper explores how different smooth and PL-structures on high-dimensional manifolds affect geometric invariants like minimal volume and Yamabe invariant, revealing that some invariants distinguish smooth structures while others do not.

Contribution

It introduces a procedure to compute key geometric invariants for high-dimensional manifolds with various smooth and PL-structures, including exotic and fake tori.

Findings

Yamabe invariant distinguishes standard from exotic smooth structures.

Minimal volume is not an invariant of smooth structures.

Fundamental group does not restrict vanishing of minimal volume or collapse.

Abstract

In this short note, exploits of constructions of $\mathcal{F}$-structures coupled with technology developed by Cheeger-Gromov and Paternain-Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature. This includes all exotic and all fake tori of dimension greater than four. We observe that the minimal volume is not an invariant of the smooth structures, yet the Yamabe invariant does discern the standard smooth structure from all the others. We also observe that the fundamental group places no restriction on the vanishing of the minimal volume and collapse with bounded sectional curvature for high-dimensional manifolds.