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Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point.

Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable spaces and $X$ has the FPP, does it follow that $Y$ also has FPP? Another way to put it: Can one force a fixed point for a self-map of a compact by a "non-homological" argument?

I do not know an answer to this even for finite simplicial complexes, but my primary interest is in locally connected finite-dimensional compacts.

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I've added the dynamical systems tag since the fixed point property is involved. Feel free to remove if you find it unsuitable. –  Vidit Nanda Sep 7 '13 at 16:19
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A very relevant (and very nice!) result is this: every compact connected CW-complex is weakly equivalent to a space with the FPP. arxiv.org/abs/1307.1722 –  Mariano Suárez-Alvarez Sep 7 '13 at 17:23
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@MarianoSuárez-Alvarez the result is definitely very nice, but it feels like cheating since the space is non-Hausdorff. –  Vidit Nanda Sep 7 '13 at 17:51
    
There are tons of such examples, and they are interesting. (I hope to dig some more to the already provided examples). –  Włodzimierz Holsztyński Sep 7 '13 at 18:12

3 Answers 3

up vote 16 down vote accepted

Lovely question! Sadly, the answer is "no" in the sense that the fixed point property is not homotopy-invariant even in the category of finite polyhedra. In fact, it is also not invariant under the operations of taking products or suspensions.

See the three page paper of W Lopez called "An example in the fixed point theory of polyhedra" for the construction of an explicit counterexample to your desired property as well as the two properties listed above. Basically, Lopez's construction involves two finite polyhedra $X$ and $Y$ whose wedge product has the fixed point property but whose union along an edge does not (!!). The Corollary to Theorem 3 on the second page of the linked pdf is of interest.

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Very nice, thank you! I was hoping for a positive answer, but alas... –  Misha Sep 7 '13 at 14:08

The title of this paper

Kinoshita, Shin'ichi, On some contractible continua without fixed point property, Fund. Math. 40, (1953). 96–98, MR0060225

gives a negative answer to your question. One of the examples is a compact cone. (Every contractible continuum is homotopy-equivalent to a one-point space.)

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Thank you, Wlodek: hard to believe that such thing is possible... –  Misha Sep 8 '13 at 6:37
    
Unsuccessful attempts at inductive proofs of Brouwer's fixed point theorem lead to such surprising counterexamples. –  Wlodek Kuperberg Sep 8 '13 at 16:37

If you are interested in "non-homological" arguments for proving fixed point theorems, you may find interesting this post, and especially the comment of Alon Amit, where I knew about this very nice paper of Milnor about a non-homological proof of the hairy ball and Brower theorems.

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