Topological properties of semialgebraic $G$-Sets

Abstract

The topological properties of semialgebraic actions of semialgebraic groups on semialgebraic sets are studied. Let $G$ be a compact semialgebraic group. We prove that every semialgebraic $G$-set with finitely many orbit types has a semialgebraic $G$-$\CW$ complex structure. Using this result, we also prove that every semialgebraic $G$-set with finitely many orbit types admits a semialgebraic $G$-embedding into some semialgebraic orthogonal representation space of $G$ for $G$ a compact semialgebraic linear group.

An affine semialgebraic $G$-set means a semialgebraic $G$-set which is semialgebraically $G$-homeomorphic to a $G$-invariant semialgebraic set in some semialgebraic representation space of $G$. Let $M$ and $N$ be affine semialgebraic $G$-sets. We find a one to one correspondence between the set of semialgebraic $G$-homotopy classes of semialgebraic $G$-maps from $M$ to $N$ and that of topological $G$-homotopy classes of continuous $G$-maps from $M$ to $N$. We also deal with the equivariant semialgebraic version of a theorem of J. H. C. Whitehead.

We also deal with semialgebraic $G$-vector bundles. It is proved that any semialgebraic $G$-vector bundle over an affine semialgebraic $G$-set has a semialgebraic classifying $G$-map. Moreover, we prove that the set of semialgebraic $G$-isomorphism classes of semialgebraic $G$-vector bundles over an affine semialgebraic $G$-set $M$ corresponds bijectively to the set of topological $G$-isomorphism classes of topological $G$-vector bundles over $M$.

Finally, we construct the equivariant Whitehead group of affine semialgebraic $G$-sets. It is shown that there is a well-defined Whitehead torsion for any $G$-homotopy equivalence between affine semialgebraic $G$-sets. We also prove the semialgebraic invariance of the Whitehead torsion. Moreover, we construct the restricti...