Retracts

Abstract

We present an example, due to Kinoshita, of a compact contractible subset of $\mathbb {R}^3$ and a continuous function from this set to itself that has no fixed point. A continuous function from a space to a subset of itself is a retraction if every point of the subset is a fixed point of the function, in which case the subset is a retract of the space. A subset of a space is a neighborhood retract if it is a retract of some neighborhood of itself. An absolute neighborhood retract (ANR) is a metric space that is a neighborhood retract of any metric space in which it is embedded as a closed subset. Various characterization results show that the class of ANR’s is large, encompassing manifolds and finite simplicial complexes, so it is a general setting that will eventually be shown to be well behaved for fixed point theory. An absolute retract (AR) is a metric space that is a retract of any metric space in which it is embedded as a closed subset. An ANR is an AR if and only if it is contractible. The domination theorem provides a sense in which a compact subset of an ANR can be arbitrarily well approximated by a finite simplicial complex.

Exercises

8.1

Let A be a retract of X.

• (a) Prove that if X is locally compact, then A is locally compact.

• (b) Prove that if X is locally path connected, then A is locally path connected.

8.2

Let C be the unit circle centered at the origin of $\mathbb {R}^2$. Prove that if f and g are continuous maps from C to itself, and f and g have the same winding number, then f and g are homotopic. (In general a homotopy invariant is complete if two maps with the same domain and range that have the same value of the invariant are necessarily homotopic.) Hopf’s theorem (Theorem 14.4) generalizes this to all dimensions.

8.3

Prove that a retract of an AR is an AR.

8.4

Let $X = \prod _{i=1}^\infty X_i$ be a countable cartesian product of metric spaces, endowed with the product topology.

• (a) Prove that if X is an AR, then each $X_i$ is an AR.

• (b) Prove that if each $X_i$ is an AR, then X is an AR.

• (c) Prove that if $X_i = \{0,1\}$ for all i, then X is not locally connected, hence not an ANR.

• (d) Prove that if each $X_i$ is an ANR and all but finitely many of the $X_i$ are AR’s, then X is an ANR. (The converse is also true; e.g., p. 93 of Borsuk 1967.)

8.5

Let X be a compact metric space, and let Y be an ANR.

• (a) By embedding Y in a suitable Banach space, prove that C(X, Y) is an ANR.

• (b) Prove that if Y is an AR, then C(X, Y) is an AR.

8.6

Let $D^m := \{\, x \in \mathbb {R}^m : \Vert x\Vert \le 1 \,\}$, let K be a compact subset of $\mathrm {int} \, D^m := \{\, x \in \mathbb {R}^m : \Vert x\Vert < 1 \,\}$, and let $\partial K := K \cap \overline{D^m \setminus K}$. Prove that if $f : K \rightarrow D^m$ is a continuous function with $f|_{\partial K} = \mathrm {Id}_{\partial K}$, then $K \subset f(K)$. (Extend f to a map ${\tilde{f}}: D^m \rightarrow D^m$ by setting ${\tilde{f}}(x) := x$ if $x \notin K$.)

8.7

Prove that if $X \subset \mathbb {R}^m$ is a compact AR, then $\mathbb {R}^m \setminus X$ does not have a connected component that is bounded. (A more challenging problem is to prove that if $X \subset \mathbb {R}^m$ is an ANR, then $\mathbb {R}^m \setminus X$ has finitely many connected components; e.g., p. 193 of Hu 1965.) Conclude that $\mathbb {R}^m \setminus X$ is connected.