# 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?

• I don't believe the closed 2-disc has the strong fixed point property. See i.stack.imgur.com/REdRa.png -- here the red region in the bottommost picture is taken to be the image of a surjective function, and the blue region is the image of another function. Suitably oriented, their graphs should not intersect. Apologies for the shoddy diagram. – Linus Hamilton Dec 22 '15 at 0:52
• @LinusHamilton thank you for the comment. may you more explain about the picture? – Ali Taghavi Dec 23 '15 at 21:46

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.

• As a consequence, any product space $X\times D$ fails to satisfy the strong FPP, since the continuous surjective self-maps $1_X\times f$ and $1_X\times g$ have disjoint graps too. – Pietro Majer Dec 24 '15 at 20:32
• $\ \ \ \$ h $\ \ \ \$ – Pietro Majer Dec 25 '15 at 12:06
• thanks for your interesting answer. may be this idea can be used for counter example for projective spaces, since they are the quotient of D_{n} by an appropriate action on the boundary. – Ali Taghavi Dec 25 '15 at 20:23
• And what about three pairwise disjoint graphs of surjective self-maps on $D$ ? More generally, given $X$ and $Y$, what is the maximum number of pairwise disjoint graphs of surjective maps $X\to Y$ ? – Pietro Majer Dec 25 '15 at 20:36
• Very interesting question. – Ali Taghavi Dec 26 '15 at 15:12

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?