# Attribution of theorem saying that inducing isomorphism on homology implies homotopy equivalence between H spaces that are CW complexes

Who was the first to prove this theorem and is there an "official" name for it?

Let $\phi:X\rightarrow Y$ be a map of H-spaces that are also CW-complexes. Assume $\phi$ induces isomorphisms on homology groups $H_{*}(-,\mathbb{Z})$. Then $\phi$ is a homotopy equivalence.

I learned about this theorem from my undergraduate topology professor who called it "Whitehead's theorem for H-spaces". The only source I have for it is V. Srinivas, Algebraic K-theory, Thm. A.53. It does not appear in any of J.H.C. Whitehead's writings, and I guess my professor called it so because of the analogy with Whitehead's theorem on weak homotopy equivalences. Does anybody know where the first proof appeared and if that name is official?

• I have been taught it as "Whitehead's theorem" too, maybe the author is George Whitehead (no relation)? Oct 20 '17 at 16:19
• Also, the theorem holds more generally for simple spaces (and even for all spaces if you extend it to homology of local systems). I refuse to consider spaces not equivalent to CW complexes ;). Oct 20 '17 at 16:40

The result for simply-connected spaces is usually attributed to J.H.C. Whitehead, in particular Theorem 14 in

• J.H.C Whitehead. Combinatorial homotopy II. Bull. Amer. Math. Soc. 55, (1949). 453–496.

The result was generalized to nilpotent spaces in

• E. Dror. A generalization of the Whitehead theorem. In: Lecture Notes in Mathematics 249 (1971), pp. 13-22.

To quote from the corresponding section on simple spaces:

If both $X$ and $Y$ are simple, then [the conditions of the Theorem] are trivially satisfied and thus $\pi_\ast f$ is an isomorphism if $H_\ast f$ is. This example includes any map between $H$-spaces. Note that in this case obstruction theory arguments or a careful relative Hurewicz argument would do.

Note that e.g. in G.W. Whitehead's book "Elements of homotopy theory", the Whitehead theorem above (for simply connected spaces) is proved as application of the relative Hurewicz theorem. So maybe there is an earlier reference, and maybe people knew before Dror's paper. But it's the earliest explicit statement I could find.