Proof of semialgebraic covering mapping cylinder conjecture with semialgebraic covering homotopy theorem

In this paper we prove the semialgebraic version of Palaisʹ covering homotopy theorem, and use this to prove Bredonʹs covering mapping cylinder conjecture positively in the semialgebraic category. Bredonʹs conjecture was originally stated in the topological category, and a topological version of our semialgebraic proof of the conjecture answers the original topological conjecture for topological G-spaces over “simplicial” mapping cylinders.