Cycles on algebraic models of smooth manifolds

Abstract

Every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of $M$. We study modulo $2$ homology classes represented by algebraic subsets of $X$, as $X$ runs through the class of all algebraic models of $M$. Our main result concerns the case where $M$ is a spin manifold.