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We study Poincar\'e type $L^p$ inequality on a compact semialgebraic subsetof $\R^n$ for $p>>1$. First we derive a local inequality by using a Lipschitzdeformation retraction with estimates on its derivatives. Then, we extend thelocal inequality to a global inequality by employing double complex technique.As a consequence we obtain an isomorphism between $L^p$ cohomology and singularcohomology of a normal compact semialgebraic pseudomanifold.

On $L^p$ inequality for differential forms and $L^p$ cohomology of a semialgebraic set for $p>>1$