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Reduced degree into the 0-sphere is homotopy invariant and multiplicative

Statement

Let $(P, s)$ be a balanced oriented finite $0$ -manifold, so that every map $P \to S^{0}$ has a reduced degree, and let $f : P \to S^{0}$ be a map (The reduced degree of a map into the 0-sphere).

(i) If $F : P \times [0, 1] \to S^{0}$ is continuous and $F_{t} (x) : = F (x, t)$ , then $F_{t} = F_{0}$ for every $t \in [0, 1]$ ; in particular $\deg F_{t} = \deg F_{0}$ .

(ii) If $g : S^{0} \to S^{0}$ is any map and $\deg g$ denotes its reduced degree as a self-map of $S^{0}$ , then $\deg (g \circ f) = \deg (g) \deg (f)$ .

Facts & Assumptions

Given: A balanced oriented finite $0$ -manifold $(P, s)$ with $\sum_{x \in P} s (x) = 0$ and a map $f : P \to S^{0}$ .

[F1]

$S^{0} = {- 1, + 1}$ is oriented by the boundary orientation of $[- 1, 1]$ : the point $+ 1$ has sign $+ 1$ and $- 1$ has sign $- 1$ . A reduced degree is defined only for a balanced source: $\sum_{x \in P} s (x) = 0$ , and then $\deg (h) = \frac{1}{2} \sum_{x \in P} s (x) h (x) \in Z$ for a map $h : P \to S^{0}$ . For $P = S^{0}$ itself this reads $\deg (h) = \frac{h (+ 1) - h (- 1)}{2} \in {- 1, 0, + 1}$ (The reduced degree of a map into the 0-sphere).

[F2]

$S^{0} = S_{2} (0, 1) \subseteq R$ carries the subspace topology (Euclidean spheres and closed balls as subspaces of $R^{n}$ , Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). The interval $[0, 1]$ is order-convex, hence connected, and a product of connected spaces is connected (A subset of $R$ is connected if and only if it is order-convex, that is, an interval, A product of connected spaces is connected in the product topology, and that argument is a theorem of ZF; for an infinite index set it is the assertion that the product of nonempty spaces is nonempty that uses the Axiom of Choice); in particular every ${x} \times [0, 1] \subseteq P \times [0, 1]$ is connected.

[F3]

In $S^{0} = {- 1, + 1} \subseteq R$ the two singletons are open for the subspace topology: ${+ 1} = S^{0} \cap B_{2} (+ 1, 1)$ and ${- 1} = S^{0} \cap B_{2} (- 1, 1)$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ , Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). A continuous map from a connected space into $S^{0}$ is therefore constant: the preimages of the two disjoint open singletons would otherwise separate the domain.

Proof

Proof technique: direct; split the composition law by the three possible reduced degrees of the self-map $g$ .

1.1F2F3algebra

The product $P \times [0, 1]$ is the disjoint union of the connected subsets ${x} \times [0, 1]$ , one for each $x \in P$ ; a continuous map from a connected space into $S^{0}$ is constant, so $F$ is constant on each ${x} \times [0, 1]$ , hence $F_{t} (x) = F (x, t) = F (x, 0) = F_{0} (x)$ for all $x \in P$ and $t \in [0, 1]$ ; equal maps have equal reduced degree, which proves (i).

1.2F1algebra

Suppose $g$ is not constant. A map $S^{0} \to S^{0}$ is determined by the pair $(g (+ 1), g (- 1))$ , so a nonconstant $g$ sends $+ 1$ and $- 1$ to different values; hence $g$ is either the identity, with $σ : = + 1$ and $g (y) = y$ for both $y$ , or the antipodal map $g (y) = - y$ , with $σ : = - 1$ , so that $g (y) = σ y$ for all $y \in S^{0}$ ; then $g \circ f = σ f$ and, by linearity of the defining signed count, $\deg (g \circ f) = \frac{1}{2} \sum_{x} s (x) σ f (x) = σ \deg (f)$ , while $σ = \deg (g)$ by [F1], the identity and the antipodal map having reduced degrees $+ 1$ and $- 1$ ; thus $\deg (g \circ f) = \deg (g) \deg (f)$ .

2.1F1step 1.2algebra∎

If $g$ is constant with value $c \in S^{0}$ , then $g \circ f$ is the constant map $c$ and $\deg (g \circ f) = \frac{1}{2} c \sum_{x \in P} s (x) = 0$ by balance, while $\deg (g) = \frac{1}{2} (c - c) = 0$ by [F1]; hence again $\deg (g \circ f) = 0 = \deg (g) \deg (f)$ , which completes (ii).

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Sources

Allen Hatcher, Algebraic Topology, §2.1 and §3.1 (standard reference, not scraped)