# Which homotopy 2-types are H-spaces?

Let $$X$$ be a connected CW-complex with $$\pi_k(X)$$ trivial for $$k >2$$. Is it known under which circumstances $$X$$ is an $$H$$-group?

I have been able only to deduce the necessary condition that $$\pi_1(X)$$ has to be abelian and act trivially on $$\pi_2(X)$$. Furthermore, if the necessary condition holds, vanishing of the Postnikov invariant $$\beta \in H^3( \pi_1(X), \pi_2(X))$$ is a sufficient condition. Thus the interesting case is that of $$\beta$$ nonzero.

• A slightly stronger and more natural condition (I don't know if it's equivalent to being an $H$-group in this case) is that $X$ has a delooping $BX$; this means not only that $\pi_1$ is abelian and acts trivially on $\pi_2$ but that the Postnikov invariant must arise from a Postnikov invariant in $H^4(B^2 \pi_1, \pi_2)$. This is exactly the space of quadratic maps $\pi_1 \to \pi_2$, although I don't know how to describe the map to $H^3(B \pi_1, \pi_2)$ in these terms. – Qiaochu Yuan Oct 7 '20 at 19:37