The generalization of Brouwer's fixed point theorem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:23:37Z http://mathoverflow.net/feeds/question/59796 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59796/the-generalization-of-brouwers-fixed-point-theorem The generalization of Brouwer's fixed point theorem? Alex Gavrilov 2011-03-28T04:05:56Z 2011-03-30T07:18:20Z <p>Let <code>$X$</code> be a contractible compact [edit: locally connected] topological space (Hausdorff and second countable). Let <code>$f\colon X\to X$</code> be a continuous map. Then (I suppose) <code>$f$</code> has a fixed point. Personally, I cannot think of a better generalization of Brouwer's fixed point theorem, but is it true?</p> http://mathoverflow.net/questions/59796/the-generalization-of-brouwers-fixed-point-theorem/59797#59797 Answer by Daniel Litt for The generalization of Brouwer's fixed point theorem? Daniel Litt 2011-03-28T04:34:34Z 2011-03-30T07:18:20Z <p>No. I believe the first counterexample is from:</p> <p>Kinoshita, S. On Some Contractible Continua without Fixed Point Property. Fund. Math. 40 (1953), 96-98</p> <p>which I unfortunately can't find online. Kinoshita's example is described on page 127 in this excellent <a href="http://www.jstor.org/stable/2317258" rel="nofollow">article by Bing</a>, however.</p> <p>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. <a href="http://www.google.com/url?sa=t&amp;source=web&amp;cd=1&amp;ved=0CBkQFjAA&amp;url=http%253A%252F%252Fmatwbn.icm.edu.pl%252Fksiazki%252Ffm%252Ffm107%252Ffm10711.pdf&amp;ei=VteSTYmMDInmsQPd_PW9Cw&amp;usg=AFQjCNH6ceZLPpJxw5lVy0sc-p_8DdudaQ&amp;sig2=Zr9Sn7SIVPRC1dpyhggXAA" rel="nofollow">here</a>; Borsuk's paper, which I can't find online, is:</p> <p>K. Borsuk, Sur un continua acyclique qui se laisse transformer topologiquement en lui-meme sans points invariants, Fund. Math. 24 (1935), pp. 51-58.</p> <p>This suggests that if the OP's conjecture is true, it is unlikely to be open to homological attacks. </p>