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?

5$\begingroup$ It is true if $X$ is triangulable by the Lefschetz fixed point theorem. I don't know what happens if $X$ is not triangulable. $\endgroup$ – Qiaochu Yuan Mar 28 '11 at 4:11

4$\begingroup$ 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. $\endgroup$ – Daniel Litt Mar 28 '11 at 23:10
No. I believe the first counterexample is from:
Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 9698
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 luimeme sans points invariants, Fund. Math. 24 (1935), pp. 5158.
This suggests that if the OP's conjecture is true, it is unlikely to be open to homological attacks.

2$\begingroup$ The original article can be found at: matwbn.icm.edu.pl/ksiazki/fm/fm40/fm4019.pdf $\endgroup$ – Michael Greinecker Mar 28 '11 at 5:35

$\begingroup$ Thank you. The example is nice! (By the way, why he call a compactum a continuum?) $\endgroup$ – Alex Gavrilov Mar 28 '11 at 11:29

$\begingroup$ Oops! I know why he call it this. What I do not know is how to delete my own foolish remark:( $\endgroup$ – Alex Gavrilov Mar 28 '11 at 12:14

$\begingroup$ I do not know if it is a right policy, but i added the local connectedness condition. Surely I missed it initially. $\endgroup$ – Alex Gavrilov Mar 28 '11 at 12:28

4$\begingroup$ @Alex: adding details to your questions is perfectly fine and should be encouraged. If you feel like you need to point out that a detail was not there before, you can always write it like: Let X be a contractible compact [edit: locally connected] topological space $\endgroup$ – Thierry Zell Mar 28 '11 at 20:16