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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
A very relevant (and very nice!) result is this: every compact connected CW-complex is weakly equivalent to a space with the FPP. – Mariano Suárez-Alvarez Sep 7 '13 at 17:23
@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

4 Answers 4

up vote 20 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.

Update (4th Oct 2015): I have also been looking for positive results lately, and one good source is Robert Brown's Handbook of Topological Fixed Point Theory (the Google book is here. Theorem 8.11 in the book is this cool result of Jerrard:

Suppose $X$ and $Y$ are compact polyhedra so that the Lefschetz numbers of every self-map $X \to X$ and $Y \to Y$ are nonzero, and so that every composite $H_n(X) \to H_n(Y) \to H_n(X)$ is trivial for $n > 0$. Then $X \times Y$ has the fixed point property.

So if you can decompose your space as a nice enough product of fixed point spaces, then there is some hope depending on their homology...

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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 $X$ is a connected polyhedron, a point $x\in X$ is said to be a global separating point if $X-\{x\}$ is not connected.

Theorem ([1], Theorem 7.1) In the category of compact connected polyhedra without global separating points, the fixed point property is a homotopy type invariant.

The example by Lopez mentioned in Vidit Nanda's answer shows that the hypothesis about global separating points is fundamental. This theorem is proved using Nielsen theory, which takes into account not only homology but also the fundamental groupoid.

Let $f:X \to X$ be a self map of a compact connected polyhedron. Two fixed points $x_0$, $x_1$ of $f$ are said to be equivalent if there is a path $c$ from $x_0$ to $x_1$ such that $c$ and $f\circ c$ are homotopic. A fixed point class of $f$ is an equivalence class of this relation (there is another definition using the universal cover of $X$ that takes into account also many empty fixed point classes). Each fixed point class has a number associated to it, its index, that measures the number of fixed points in that class. The sum of these indices, taken over the set of fixed point classes is the Lefschetz number of $f$ (Lefschetz-Hopf Theorem). The Nielsen number of $f$, $N(f)$ is the number of fixed point classes having nonzero fixed point index. The number $N(f)$ is a homotopy invariant of $f$ and therefore is a lower bound for the number of fixed points of any map homotopic to $f$. With some local hypotheses (see [1], Main Theorem) there is a map $g$ homotopic to $f$ having exactly $N(f)$ fixed points. To prove this, we can assume that $f$ has isolated fixed points (this goes back to Hopf). The local hypotheses are then used to combine two equivalent fixed points. Some references for this beautiful subject are [2] and [3].

[1] B. J. Jiang. On the least number of fixed points. Amer. J. Math., 102 (1980), 749-763.

[2] B. J. Jiang. A primer of Nielsen fixed point theory. Handbook of topological fixed point theory, 617--645, Springer, Dordrecht, 2005.

[3] B. J. Jiang. Lectures on Nielsen fixed point theory. Contemporary Mathematics, 14. American Mathematical Society, Providence, R.I., 1983. vii+110 pp.

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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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