A property stronger than the fixed point property Assume that $X$ is  a topological space. We say that $X$ satisfies the strong  fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-map on $X$. For example the interval $I=[0,1]$ satisfy this property.
An equivalent definition: If two continuous  self-maps $f,g$ on $X$ have non intersecting graphs, then neither $f$ nor $g$ is  surjective. 
Are there  some examples of manifolds (with or without boundary) of higher dimension with this property? In particular, do the closed $2$_disc, or the even dimensional real or complex projective spaces satisfy this property?
 A: Here is an example of two surjective continuous  self-maps $f$ and $g$ on the closed unit disk  whose graphs are disjoint. 
Consider the surjective continuous maps $F(x,y):=x e^{2i\pi y }$ and $G(x,y):=(x-1)e^{2i\pi y }$ from the closed unit square $[0,1]^2$ to the closed unit disk $D$. Clearly we have $F(x,y)=G(x,y)$ at no point $(x,y)\in [0,1]^2$. If we pre-compose $F$ and $G$ with a homeomorphism $D\to [0,1]^2$, we get the self-maps $f$ and $g$ as claimed.
A: Three related references are the following. 
If $f:D^2\to D^2$ is such that the map $H^2(D^2,S^1)\to H^2(D^2,f^{-1}(S^1))$ induced in Cech cohomology is non-trivial, then the graph of any other map $g:D^2\to D^2$ intersects the graph of $f$. This is part of Theorem A in "W. Holsztynski, On the product and composition of universal mappings of manifolds into cubes, Proc Amer. Math. Soc, 58(1976), 311–314".
If $f,g:D^2\to D^2$ are commuting maps, do their graphs intersect? this question appears here: Two commuting mappings in the disk
If $f,g:D^2\to D^2$ are commuting homeomorphisms, do the graphs of $f,g$ and of the identity $1_{D^2}$ intersect? This question has been asked here: Do commuting homeomorphisms of the $2$-disk have a common fixed point?
