Topological Consequences

Abstract

This chapter presents topological consequences of the degree and fixed point index. The Euler characteristic of a compact ANR is the index of the space’s identity function. For a triangulated 2-manifold this is shown to agree with Euler’s formula $F - E + V$. The Lefschetz fixed point theorem can be understood as asserting that an upper hemicontinuous contractible valued correspondence from a compact ANR to itself has a fixed point if its index does not vanish. The Hopf theorem asserts that if two functions from a sphere to itself have the same degree, then they are homotopic. Several results study the fixed points of a map from a sphere to itself, and the fixed points of various compositions of such a map with the map taking each point in the sphere to its antipode. A map between spheres is antipodal if the image of the antipode of each point in the domain is the antipode of the point’s image. The degree of an antipodal map from a sphere to itself is odd. This result implies the Borsuk–Ulam theorem and invariance of domain, which asserts that if a function from an open subset of $\mathbb {R}^m$ to $\mathbb {R}^m$ is continuous and injective, then it is an embedding and its image is open. For an upper hemicontinuous convex valued correspondence, if a connected set of fixed points has index zero, then it is inessential.

Exercises

14.1

Prove that there is no injective continuous function $f : S^n \rightarrow \mathbb {R}^n$, where $S^n := \{\, x \in \mathbb {R}^{n+1} : \Vert x \Vert = 1 \,\}$.

14.2

Suppose that X is a topological space and $A \subset X$.

• (a) Prove that if (X, A) has the homotopy extension property, then A is a closed neighborhood retract in X.

• (b) Use Proposition 8.13 to show that if X is an ANR and A is a closed subset of X, then (X, A) has the homotopy extension property.

14.3

(This problems presumes elementary concepts of group theory.) Let $I^n := [0,1]^n$, and let $\partial I^n$ be its boundary. We think of $I^n$ with $\partial I^n$ collapsed to a point as a representation of the n-sphere $S^n := \{\, x \in \mathbb {R}^{n+1} : \Vert x\Vert = 1 \,\}$. For pairs (X, A) and (Y, B), where X and Y are topological spaces and $A \subset X$ and $B \subset Y$, a continuous function from (X, A) to (Y, B) is a continuous $f : X \rightarrow Y$ such that $f(A) \subset B$, and a homotopy of such functions is a continuous $h : X \times [0,1] \rightarrow Y$ such that $h(A \times [0,1]) \subset B$. For a space X with a basepoint $x_0$ we write $(X, x_0)$ rather than $(X,\{x_0\})$. Let $\pi _n(X, x_0)$ be the set of homotopy classes [f] of maps $f : (I^n,\partial I^n) \rightarrow (X, x_0)$. For $[f], [g] \in \pi _n(X, x_0)$ let $[f] * [g]$ be the homotopy class of the map

$$(t_1,\ldots , t_n) \mapsto {\left\{ \begin{array}{ll} f(2t_1,t_2, \ldots , t_n) , &{} t_1 \le 1/2, \\ g(2t_1 - 1,t_2, \ldots , t_n), &{} 1/2 \le t_1 \;. \end{array}\right. }$$

• (a) Prove that $[f] * [g]$ is well defined in the sense of not depending on the choice of representatives f and g.

• (b) Prove that $*$ is a group operation:

  • (i) $*$ is associative.

  • (ii) If $e : I^n \rightarrow X$ is the constant map with value $x_0$, then $[e] * [f] = [f] = [f] * [e]$.

  • (iii) If $f^{-1}$ is the map $(t_1, \ldots , t_n) \mapsto f(1 - t_1,t_2, \ldots , t_n)$ then $[f] * [f^{-1}] = [e] = [f^{-1}] *[f]$.

• (c) Explain why $\pi _n(X, x_0)$ is abelian if $n \ge 2$. (Instead of trying to define a particular homotopy, draw some pictures.)

The groups $\pi _n(X, x_0)$ are called the homotopy groups of $(X, x_0)$, and $\pi _1(X, x_0)$ is the fundamental group of $(X, x_0)$. Let $m : (X, x_0) \rightarrow (Y, y_0)$ be continuous, and let $\pi _n(m) : \pi _n(X, x_0) \rightarrow \pi _n(Y, y_0)$ be the function $[f] \mapsto [m \circ f]$. (This is evidently well defined, in the sense of independence of the choice of representative.) Often we write $m_*$ rather than $\pi _n(m)$.

• (d) Prove that $\pi _n(m)$ is a homomorphism.

• (e) Prove that $\pi _n$ is a covariant functor from the category of pointed spaces and continuous maps to the category of groups and homomorphisms.

• (f) If $(M, p_0)$ is an n-dimensional oriented manifold with basepoint $p_0$, interpret the degree of maps $S^n \rightarrow M$ as a homomorphism from $\pi _n(M, p_0)$ to $\mathbb {Z}$. What does Hopf’s theorem say about this homomorphism when $M = S^n$?

We identify $I^{n-1}$ with $\{\, t \in I^n : t_n = 0 \,\}$, and we let $J^{n-1}$ be the closure of $I^n \setminus I^{n-1}$. We think of $(I^n,\partial I^n)$ with $J^{n-1}$ collapsed to a point as a representation of the pair $(D^n, S^{n-1})$ where $D^n$ is the n-disk $D^n := \{\, x \in \mathbb {R}^n : \Vert x\Vert \le 1 \,\}$. A topological triple is a triple (X, A, B) where X is a topological space and $B \subset A \subset X$. We define continuous maps between triples and homotopies of such maps as above. If X is a topological space and $x_0 \in A \subset X$, for $n \ge 2$ let $\pi _n(X,A, x_0)$ be the set of homotopy classes of maps $f : (I^n, \partial I^n, J^{n-1}) \rightarrow (X,A, x_0)$.

• (g) Discuss how to modify of the arguments above to show that the binary operation $*$ (defined just as above) makes $\pi _n(X,A, x_0)$ into a group, and that a continuous map $m : (X,A, x_0) \rightarrow (Y,B, y_0)$ induces a homomorphism $m_* : \pi _n(X,A, x_0) \rightarrow \pi _n(Y,B, y_0)$.

• (h) In the sequence

  $$\cdots \rightarrow \pi _2(A, x_0) \rightarrow \pi _2(X, x_0) \rightarrow \pi _2(X,A, x_0) \rightarrow \pi _1(A, x_0) \rightarrow \pi _1(X, x_0)$$

  the homomorphisms $i_* : \pi _n(A, x_0) \rightarrow \pi _n(X, x_0)$ and $j_* : \pi _n(X, x_0) \rightarrow \pi _n(X,A, x_0)$ are induced by the inclusions $i : (A, x_0) \rightarrow (X, x_0)$ and $j : (X, x_0,x_0) \rightarrow (X,A, x_0)$, and the homomorphism $\partial : \pi _n(X,A, x_0) \rightarrow \pi _{n-1}(A, x_0)$ is $[f] \mapsto [f|_{I^{n-1}}]$. (You should convince yourself that $\partial$ is a homomorphism.) Prove that the sequence is exact: the image of each homomorphism (except the last) is the kernel of its successor.

• (i) Prove that for all $d \ge 1$ and $n \ge 2$, $\partial : \pi _n(D^d, S^{d-1}, p_0) \rightarrow \pi _{n-1}(S^d, p_0)$ is an isomorphism. (Of course $p_0$ is an arbitrary point in $S^{d-1}$.)

14.4

Recall that for $n \ge 0$, n-dimensional (real) projective space $P^n$ is the set of 1-dimensional subspaces of $\mathbb {R}^{n+1}$, topologized in the “obvious” way. Alternatively, we may regard $P^n$ as the space obtained from $S^n$ by identifying antipodal points, so there is a canonical map $b_n : S^n \rightarrow P^n$ taking x to $\{x,-x\}$.

• (a) Show that $P^n$ is orientable if and only if n is odd.

• (b) Show that $\mathrm {Id}_{P^n}$ is not homotopic to a constant map.

• (c) Prove that if $\gamma : [0,1] \rightarrow P^n$ is continuous and $b_n(x_0) = \gamma (0)$, then there is a unique continuous ${\tilde{\gamma }}: [0,1] \rightarrow S^n$ such that ${\tilde{\gamma }}(0) = x_0$ and $b_n \circ {\tilde{\gamma }}= \gamma$.

If ${\tilde{\gamma }}: [0,1] \rightarrow S^n$ is continuous, $\gamma = b_n \circ {\tilde{\gamma }}$, $\gamma (0) = \gamma (1)$, and ${\tilde{\gamma }}(1) = a_n({\tilde{\gamma }}(0))$, then we say that $\gamma$ is an antipodal loop.

• (d) For a continuous $f : P^n \rightarrow P^n$, prove that there is a continuous ${\tilde{f}}: S^n \rightarrow S^n$ such that $f \circ b_n = b_n \circ {\tilde{f}}$ if and only if $f \circ \gamma$ is an antipodal loop whenever $\gamma : [0,1] \rightarrow P^n$ is an antipodal loop.

• (e) For an arbitrary $p_0 \in P^n$, prove that $\pi _1(P^n, p_0)$ is the group of integers mod 2.