Cobordism, spin structures, and profinite completions

TL;DR

This paper demonstrates that for aspherical manifolds with isomorphic profinite completions of their fundamental groups, the manifolds are cobordant with matching signatures modulo 8, and their spin structures are equivalent.

Contribution

It establishes a link between profinite completions of fundamental groups and geometric-topological properties like cobordism and spin structures for aspherical manifolds.

Findings

Manifolds with isomorphic profinite completions are cobordant.

Signatures of such manifolds agree modulo 8.

Spin and spin^C structures are equivalent under profinite isomorphism.

Abstract

Let $M$ and $N$ be smooth closed connected aspherical manifolds with good (in the sense of Serre) fundamental groups $G$ and $H$. We show that if $\widehat G\cong \widehat H$, then $M$ and $N$ are cobordant and the signatures of $M$ and $N$ agree modulo $8$. Moreover, $M$ is spin (resp.spin$^\CC$) if and only if $N$ is spin (resp.spin$^\CC$). We consider some analogous results for compact connected aspherical manifolds.