A manifold $X$ has the fixed-point property if for every continuous map $f:X→X$ there is $x∈X$ with $f(x)=x$. Examples of such spaces are disks and the real projective plane $\mathbb{RP}^2$.


**Question:** If a compact manifold $X$ has the fixed-point property, does $X\times X$ necessarily have the fixed-point property?


**Known:** 


False for $X$ a polyhedron. See [An example in the fixed point theory of polyhedra. Bull. Amer. Math. Soc. 73(1967), 922-924](https://projecteuclid.org/download/pdf_1/euclid.bams/1183529109).


False for various other spaces. See [MO question 283930](https://mathoverflow.net/questions/283930/squaring-a-space-with-the-fixed-point-property).


True for compact surfaces: the only such surface with the fixed-point property is $\mathbb{RP}^2$, and $\mathbb{RP}^2\times \mathbb{RP}^2$ also has the fixed-point property, by the Lefschetz fixed point lemma.



Compact manifold $X$ having fixed-point property but $X\times X$ does not
Not a full answer, but the answer seems to not be known. The question is open for closed manifolds (see the mathscinet [review](https://mathscinet.ams.org/mathscinet-getitem?mr=3584128) of Kwasik and Sun's [paper](https://arxiv.org/pdf/1609.05802.pdf), MR3584128). The answer is yes for other manifolds like $\mathbb{CP}^2$, but no for the *symmetric product* of closed manifolds according to the same paper (Theorem 2). One can take $X=\mathbb{RP}^4\#\mathbb{RP}^4\#\mathbb{RP}^4$ which has the fpp, but $X(2)=(X\times X)/\mathbb{Z}\_2$ does not. 


