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Let $X$ be a contractible compact [edit: locally connected] topological space (Hausdorff and second countable). Let $f\colon X\to X$ be a continuous map. Then (I suppose) $f$ has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?

edited Mar 28 2011 at 22:37

asked Mar 28 2011 at 4:05

Alex Gavrilov
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It is true if $X$ is triangulable by the Lefschetz fixed point theorem. I don't know what happens if $X$ is not triangulable. – Qiaochu Yuan Mar 28 2011 at 4:11

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Alex: it is generally considered to be bad etiquette to edit your question substantively after answers have been posted (and accepted!). It might be worth reading some of the literature (e.g. Bing's article) and then posting a new question, with whatever information you discover upon doing so. – Daniel Litt Mar 28 2011 at 23:10

Daniel Litt · Accepted Answer · 2011-03-30 07:18:20Z UTC

No. I believe the first counterexample is from:

Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 96-98

which I unfortunately can't find online. Kinoshita's example is described on page 127 in this excellent article by Bing, however.

EDIT: The question has been revised to add the local connectivity condition; as stated, I think the question is open. If "contractible" is replaced with "acyclic" there are counterexamples dating back to Borsuk, referenced e.g. here; Borsuk's paper, which I can't find online, is:

K. Borsuk, Sur un continua acyclique qui se laisse transformer topologiquement en lui-meme sans points invariants, Fund. Math. 24 (1935), pp. 51-58.

This suggests that if the OP's conjecture is true, it is unlikely to be open to homological attacks.

The generalization of Brouwer’s fixed point theorem?

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