${\mathbb{Z}}_k$-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 $H_{\ast }(B{\mathbb{Z}}_p;{\mathbb{Z}}_p)$ using ${\mathbb{Z}}_p$-stratifolds.