I've just started reading J.P. May's book The Geometry of Iterated Loop Spaces and am misunderstanding something. Somewhere, it's asserted that if an H-space X can be delooped twice, the its multiplication is homotopy commutative. Should the equivalence $X \simeq \Omega^2 Y$ should preserve the H-space structure for this to hold?

If not, then I don't understand something about H-spaces or loop spaces, since it seems that $\Omega^2 S^1 \simeq \mathbb{Z}$, and so an H-space structure is just a map $\mu:\mathbb{Z}^2 \rightarrow \mathbb{Z}$ for which 0 is a left and right identity. Commuting up to homotopy here is just commuting, and I can write down lots of maps $\mu$ that don't commute. I would appreciate clarification!

  • 2
    $\begingroup$ Yes, the equivalence should be as $H$-spaces. Your example is problematic, as $S^1 \simeq \Omega \mathbb CP^\infty$, and $\Omega S^1 \simeq \mathbb Z$. Both equivalences are as $H$ spaces, where $S^1$ and $\mathbb Z$ are given their usual abelian group structures, and the loop spaces have a "canonical" $H$-space structure. $\endgroup$ – Justin Young Nov 4 '13 at 17:25

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