Does there exist a simply connected, non-contractible manifold $M$, which is an $H$-space, and whose rational homology groups vanish in positive degrees?

My space $M$ is in fact homotopy equivalent to the loop space of some other topological space. It seems that plenty of results about such spaces are available under the assumption that the homology is finitely generated, an assumptions that my space $M$ does not satisfy, a priori.