Classification of the crossed product $C(M){\times }_{\theta }{\mathbb{Z}}_p$ for certain pairs $(M,\theta )$

Abstract

Let $M$ be a separable compact Hausdorff space with $\dim M\le 2$ and $\theta :M\to M$ be a homeomorphism with prime period $p$ ($p\ge 2$). Set $M_{\theta }=\{x\in M|\theta (x)=x\}\ne \varnothing$ and $M_0=M\backslash M_{\theta }$. Suppose that $M_0$ is dense in $M$ and ${\mathrm{H}}^2(M_0/\theta ,\mathbb{Z})\cong 0$, ${\mathrm{H}}^2(\chi (M_0/\theta ),\mathbb{Z})\cong 0$. Let $M^{\prime }$ be another separable compact Hausdorff space with $\dim M^{\prime }\le 2$ and ${\theta }^{\prime }$ be the self--homeomorphism of $M^{\prime }$ with prime period $p$. Suppose that $M_0^{\prime }=M^{\prime }\backslash M_{{\theta }^{\prime }}^{\prime }$ is dense in $M^{\prime }$. Then $C(M){\times }_{\theta }{\mathbb{Z}}_p\cong C(M^{\prime }){\times }_{{\theta }^{\prime }}{\mathbb{Z}}_p$ iff there is a homeomorphism $F$ from $M/\theta$ onto $M^{\prime }/{\theta }^{\prime }$ such that $F(M_{\theta })=M_{{\theta }^{\prime }}^{\prime }$. Thus, if $(M,\theta )$ and $(M^{\prime },{\theta }^{\prime })$ are orbit equivalent, then $C(M){\times }_{\theta }{\mathbb{Z}}_p\cong C(M^{\prime }){\times }_{{\theta }^{\prime }}{\mathbb{Z}}_p$.

arXiv comment11 pages. Filomat 2010 (to appear)