$\mathbb{Z}_k$-stratifolds

TL;DR

This paper introduces $rac{k}$-stratifolds, a geometric framework that represents all homology classes with $rac{k}$ coefficients, extending previous concepts and providing new interpretations of spectral sequences and homology classes.

Contribution

It defines $rac{k}$-stratifolds and demonstrates their sufficiency in representing homology classes, offering geometric insights into spectral sequences and explicit representatives for homology of classifying spaces.

Findings

$rac{k}$-stratifolds can represent all $rac{k}$-homology classes.

Provides geometric interpretation of Bockstein sequences and Atiyah-Hirzebruch spectral sequence.

Constructs explicit geometric representatives for $H_*(Brac{p}{p})$ using $rac{p}{p}$-stratifolds.

Abstract

Generalizing the ideas of $\mathbb{Z}_k$-manifolds from Sullivan and stratifolds from Kreck, we define $\mathbb{Z}_k$-stratifolds. We show that the bordism theory of $\mathbb{Z}_k$-stratifolds is sufficient to represent all homology classes of a $CW$-complex with coefficients in $\mathbb{Z}_k$. We present a geometric interpretation of the Bockstein long exact sequences and the Atiyah-Hirzebruch spectral sequence for $\mathbb{Z}_k$-bordism ($k$ an odd number). Finally, for $p$ an odd prime, we give geometric representatives of all classes in $H_*(B\mathbb{Z}_p;\mathbb{Z}_p)$ using $\mathbb{Z}_p$-stratifolds.