(\mathbb{Z}_k\)-stratifolds

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.