Fixed-point property

A mathematical object $X$ has the fixed-point property if every suitably well-behaved mapping from $X$ to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set $P$ is said to have the fixed point property if every increasing function on $P$ has a fixed point.

Definition

Let $A$ be an object in the concrete category $\mathbf {C}$. Then $A$ has the fixed-point property if every morphism (i.e., every function) $f:A\to A$ has a fixed point.

The most common usage is when $\mathbf {C} =\mathbf {Top}$ is the category of topological spaces. Then a topological space $X$ has the fixed-point property if every continuous map $f:X\to X$ has a fixed point.

Examples

Singletons

In the category of sets, the objects with the fixed-point property are precisely the singletons.

The closed interval

The closed interval $[0,1]$ has the fixed point property: Let $f:[0,1]\to [0,1]$ be a continuous mapping. If $f(0)=0$ or $f(1)=1$, then our mapping has a fixed point at 0 or 1. If not, then $f(0)>0$ and $f(1)-1<0$. Thus the function $f(g)=f(x)-x$ is a continuous real valued function which is positive at $x=0$ and negative at $x=1$. By the intermediate value theorem, there is some point $x_{0}$ with $g(x_{0})=0$, which is to say that $f(x)-x=0$, and so $x_{0}$ is a fixed point.

The open interval does not have the fixed-point property. The mapping $f(x)=x^{2}$ has no fixed point on the interval $(0,1)$.

The closed disc

The closed interval is a special case of the closed disc, which in any finite dimension has the fixed-point property by the Brouwer fixed-point theorem.

Topology

A retract $A$ of a space $X$ with the fixed-point property also has the fixed-point property. This is because if $r:X\to A$ is a retraction and $f:A\to A$ is any continuous function, then the composition $i\circ f\circ r:X\to X$ (where $i:A\to X$ is inclusion) has a fixed point. That is, there is $x\in A$ such that $f\circ r(x)=x$. Since $x\in A$ we have that $r(x)=x$ and therefore $f(x)=x.$

A topological space has the fixed-point property if and only if its identity map is universal.

A product of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval.

The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.

According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.

References

• Samuel Eilenberg, Norman Steenrod (1952). Foundations of Algebraic Topology. Princeton University Press.
• Schröder, Bernd (2002). Ordered Sets. Birkhäuser Boston.