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. 



fixed point property for maps of compacts
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](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183529109) 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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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](https://books.google.com/books?id=uWjYVrsw76gC&pg=PA230&lpg=PA230&dq=jerrard%20fixed%20point&source=bl&ots=as-shTLTpx&sig=-O1pvZLroYkEwlSX05-Prmcua48&hl=en&sa=X&ved=0CDAQ6AEwAmoVChMImq7xjbKpyAIVhVoeCh3rbwDL#v=onepage&q=jerrard%20fixed%20point&f=false). Theorem 8.11 in the book is this cool result of Jerrard:



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> 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.
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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...


