Algebraic and geometric models for $H_{0}$-spaces

Abstract:

For every ${H_0}$-space (i.e. a space whose rationalization is an $H$-space) we construct a space $J$ depending only on ${H^\ast }(X;{\mathbf {Z}})$ and a rational homotopy equivalence $J \to X$ (i.e. $J$ is a universal space to the left of all ${H_0}$-spaces having the same integral cohomology ring as $X$ is constructed generalizing the James reduced product. We study also the integral cohomology of ${H_0}$-spaces and we prove that under certain conditions it contains an algebra with divided powers.