Analyticity of homology classes

Abstract:

Let $W$ be a real analytic manifold and $\{ \alpha \} \in {H_p}(W,{Z_2})$. We shall say that $\{ \alpha \}$ is analytic if there exists a compact analytic subset $S$ of $W$, such that: $\{ \alpha \}= {}$ fundamental class of $S\}$. The purpose of this short paper is to prove Theorem 1. Let $W$ be a paracompact real analytic manifold; then any homology class $\{ \alpha \} \in {H_p}(W,{Z_2})$ is analytic. We remember that a similar result does not hold in the real algebraic case (see [1]).