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 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 C = 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] → [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 g(x) = 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 x₀ with g(x₀) = 0, which is to say that f(x₀) − x₀ = 0, and so x₀ is a fixed point.

The open interval does not have the fixed-point property. The mapping f(x) = x² 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 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.