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In 1966, Bob Stong published the paper Finite Topological Spaces in Trans. Amer. Math. Soc. where he gave the poset interpretation of a finite space and provided some very useful machinery to study them in this regard. [These techniques were still in use by people like JP May a decade ago, if not more recently.] One of the things Stong was able to prove using his techniques was that a finite and connected H-space is necessarily contractible. (By a finite space we mean a topological space with a finite number of points, not in the sense of "finite" CW-complexes, which is something different.) The proof is several pages long and non-trivial.

While mucking around with the results in this paper, I found that it is not too hard to prove the analogous result for co-H-spaces: a finite and connected co-H-space must be contractible. However, this is never mentioned in Stong's paper. Moreover, Stong's proof does not dualize to prove the co-H-space version. ("My" proof of the co-H-version is at least an order of magnitude simpler than the H-version, and it likewise does not dualize to prove the H-space version.)

This leads to my question. Is there some obvious and more basic (perhaps cohomological) reason that a finite and connected co-H-space must be contractible? My hunch is that there is, and this is why this is never worthy of mention in Stong's article. I feel this must have been known by other, simpler means, but I don't see what this might be.

I know that a co-H-space $X$ must have $\pi_1(X)$ a free group, but this doesn't help. After all, a paper of McCord that came out at the same time as Bob's shows that finite spaces are every bit as homotopically complicated as spaces in general. In particular, there are easily constructed finite spaces with free fundamental groups. Beyond this useless bit, I do not know of a more straightforward path to the result other than Stong's machinery of posets and cores.

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  • $\begingroup$ It is not necessary to assume connectedness in the co-H-space case (indeed, it is easy to see that any co-H-space is automatically connected). $\endgroup$ Commented Jul 12, 2023 at 22:20

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Suppose $X$ is a finite non-contractible co-H-space with structure map $t:X\to X\vee X$, with $p_0,p_1:X\vee X\to X$ the projections onto the summands. Let $Y$ be a pointed subspace of $X$ of minimal cardinality such that the inclusion map $i:Y\to X$ is not nullhomotopic. Note then that $p_0ti$ and $p_1ti$ are both non-nullhomotopic, which by minimality of $Y$ means their images must have cardinality $|Y|$. But this is impossible, since the image of $ti$ must contain points besides the basepoint which will then get collapsed to the basepoint by $p_0$ or $p_1$.

(The same argument also shows that any finite weak co-H-space (meaning the compositions $p_0t$ and $p_1t$ only have to be weakly homotopic to the identity, i.e. if you lift them to a weakly equivalent CW-complex they become homotopic) must be weakly contractible.)

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