Cobordism, spin structures, and profinite completions

TL;DR

This work establishes that profinite completions of fundamental groups of aspherical manifolds with good groups encode strong topological constraints: cobordism class, signatures modulo $8$, and the existence of spin or spin$^\mathbb{C}$ structures. The authors develop a comprehensive framework linking profinite cohomology, Steenrod operations, Wu classes, and quadratic form theory to detect Stiefel–Whitney, Pontryagin, and intersection-form data. Key contributions include proving invariance of SW classes, SW numbers, spin structures, Pontryagin classes modulo $3$, and the signature modulo $8$ under profinite equivalence, as well as providing a limiting example illustrating the necessity of goodness. The results substantially advance profinite rigidity in geometry and topology, showing that finite quotients of fundamental groups control meaningful manifold invariants for a wide class of aspherical spaces. The methods blend algebraic topology with profinite methods to bridge discrete and profinite worlds, with potential implications for classification problems in high-dimensional geometry and arithmetic groups.

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.

Theorems & Definitions (64)

• Theorem A
• Remark 1.1
• Definition 2.1: Stiefel--Whitney classes of a space
• Definition 2.2: Stiefel--Whitney classes of a manifold
• Definition 2.3
• Theorem 2.4: Wu
• Definition 2.5: Stiefel--Whitney classes of a Poincaré complex
• Definition 2.6: Stiefel--Whitney numbers
• Definition 2.7: Integral Stiefel--Whitney classes
• Remark 2.8