Homology Classes of Semi-Algebraic Sets and Mass Minimization

We associate to any compact semi-algebraic set $X\subset {\mathbb{R}}^n$ a chain complex of currents $S_{\ast }(X)$ generated by integration along semi-algebraic submanifolds and we analyze the corresponding homology groups. In particular, we show that these homology groups satisfy the Eilenberg-Steenrod axioms and further, that they are isomorphic to both the ordinary singular homology groups of $X$ and to the homology groups generated by the integral currents supported on $X$. Using this result and a certain neighborhood of $X$, we are able to prove homological mass minimization for integral currents supported on $X$, and verify that any cycle of $X$ that has sufficiently small mass is a boundary.