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I was reading the paper "Homology fibration and group completion theorem" by Segal and McDuff. During the proof of Proposition 6 (of local-to-global flavor) he states the existence of spaces $E_\mathfrak{B}$, $|\mathfrak{B}|$ homotopy equivalent to the starting spaces $E, B$.

From what I seem to be understanding, and here comes the actual question, it seems to me that the following fact is true (and unproven in the paper and its references with this generality):

Let $B$ be a paracompact, locally contractible and contractible space and let $\mathfrak{B}$ be a basis of contractible open sets. $\mathfrak{B}$ can be seen as a category with the basis open sets as objects and their inclusions as morphisms. Then its geometric realization (or precisely, the realization of its nerve) is homotopic to $B$, and hence contractible.

I have come across other lemmas similar to this where you can take any cover of contractible sets but they all use some kind of hypothesis on the finite intersections of the open sets: all of them have to be contractible. However I can't manage to prove "their" version.

In particular I can't understand how the fact that $\mathfrak{B}$ is a basis should be fundamental (I seem to have found that the equivalence is false if I replace $\mathfrak{B}$ with a generic open cover without hypothesis on the intersections: just think about covering $S^1$ with two enlarged semicircles).

Maybe some of you can quote a reference for this, or even better know a way to avoid it. Thanks

Arnav Das
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Alessandro Fenu
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    does 4G.2 answer your question? It's not stated in terms of geometric realizations but it's pretty similar. You need partitions of unity. – Andres Mejia Sep 14 '24 at 01:55
  • @AndresMejia it does not quite answer the question (however it is another way of proving the other homotopy equivalence he states, namely that of $E_\mathfrak{B}\rightarrow E$). I think ultimately that the space he is looking at is not the realization of a basis but the realization of the disintegration of $B$ with respect to the basis (that may suggests that the lemma of my question be false) – Alessandro Fenu Sep 15 '24 at 10:45

1 Answers1

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This question deserves a clear answer, if only for posterity. The McDuff-Segal paper, though ingenious and highly influential, contains some errors, one of which you have stumbled upon. This particular error was remedied by McDuff in her paper 'The homology of some groups of diffeomorphisms' (see Lemma 3.1). A modern and more careful (though also less concise as a result) exposition of the McDuff-Segal strategy is given in Miller-Palmer's 'A twisted homology fibration criterion and the twisted group-completion theorem'.

PS: If you only care about the integral homology version of the group completion theorem, this strategy can be avoided and a different one can be used, see Appendix D of Hatcher's 'A Short Exposition of the Madsen-Weiss Theorem'.

Thorgott
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