Studies on compact and noncompact semialgebraic transformation groups

Abstract

Various properties of semialgebraic actions including noncompact case are studied. Let $G$ be a semialgebraic group and $M$ a proper semialgebraic $G$-set. We prove that every point of $M$ has a semialgebraic slice and $M$ can be covered by a finite number of $G$-tubes. Using this, we obtain some pleasant results. We prove that $M$ can be embedded in a $G$-representation space if $G$ is a semialgebraic linear group. Semialgebraic version of the covering homotopy theorem is proved when $G$ is compact. With this, a conjecture introduced by Bredon is completely solved in that semialgebraic category which covers almost all reasonable topological cases. We also show that every proper semialgebraic $G$-set has a semialgebraic $G$-cell decomposition. And finally we introduce the theory of semialgebraic $G$-vector bundles and we show that every semialgebraic $G$-vector bundles over a semialgebraic set is one to one correspondence with topological $G$-vector bundles.