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Sphere self-maps of degree $\pm 1$ are exactly the homotopy equivalences

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For $m \geq 1$ , a continuous self-map $f : S^{m} \to S^{m}$ is a homotopy equivalence if and only if $∣ \deg (f) ∣ = 1$ .

Facts & Assumptions

[L1]

The ambient reflection has determinant $- 1$ and sends the outward normal $x$ at $x \in S^{m}$ to the outward normal $R (x)$ . Consequently it reverses the tangent orientation defined by placing that normal first; it is a smooth involution, hence an orientation-reversing diffeomorphism. The general diffeomorphism-degree theorem gives degree $- 1$ . (Induced boundary orientation, Degree of an orientation-preserving or reversing diffeomorphism).

Given: ${A C}_{ω}$ , an integer $m \geq 1$ and a continuous self-map $f : S^{m} \to S^{m}$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ , Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

[F1]

Sphere self-maps are homotopic exactly when their degrees agree, and every integer occurs as a degree (Sphere self-maps are homotopic exactly when their degrees agree).

[F2]

Degree is multiplicative under composition of proper smooth maps between oriented closed manifolds and the identity has degree $1$ ; for continuous sphere self-maps, homotopic maps have equal degree and $\deg (g \circ h) = \deg (g) \deg (h)$ (Degree is multiplicative under composition, Degree is homotopy invariant and multiplicative under composition, Degree of a proper smooth map by compact-support cohomology).

[F3]

The coordinate reflection $R : S^{m} \to S^{m}$ is an orientation-reversing diffeomorphism and has degree $- 1$ , and any orientation-reversing diffeomorphism between connected oriented boundaryless manifolds has degree $- 1$ (the local calculation, Degree of an orientation-preserving or reversing diffeomorphism).

[F4]

A map homotopic to a homotopy equivalence is a homotopy equivalence, and the identity is a homotopy equivalence (A continuous map homotopic to a homotopy equivalence is itself a homotopy equivalence, Homotopy equivalences, homotopy inverses and spaces of the same homotopy type).

Proof

technique · direct

1.1F2F4given

(Necessity.) Suppose $f$ is a homotopy equivalence with homotopy inverse $g$ . Then $g \circ f ≃ {id}_{S^{m}}$ , so by [F2] applied to the continuous maps, $\deg (g) \deg (f) = \deg (id) = 1$ in $Z$ ; hence $\deg (f)$ is a unit of $Z$ , that is $\deg (f) = \pm 1$ .

1.2F1F4

(Degree $1$ .) If $\deg (f) = 1 = \deg (id)$ , then [F1] gives $f ≃ {id}_{S^{m}}$ , and since the identity is a homotopy equivalence, [F4] makes $f$ a homotopy equivalence.

1.3L1F1F3F4

(Degree $- 1$ .) If $\deg (f) = - 1$ , then $\deg (f) = \deg (R)$ for the coordinate reflection $R$ of [F3], whose orientation reversal and degree are computed in [L1], so [F1] gives $f ≃ R$ ; the reflection is a diffeomorphism and hence a homotopy equivalence, so [F4] makes $f$ a homotopy equivalence.

2.1F1F4step 1.1step 1.2step 1.3∎

Steps 1.1, 1.2 and 1.3 prove both implications, so a continuous self-map of $S^{m}$ is a homotopy equivalence exactly when its degree is $\pm 1$ .

Depends on

Used by

A sphere reflection has degree minus one Example

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Victor Guillemin and Alan Pollack, Differential Topology (standard reference, not scraped)
John Milnor, Topology from the Differentiable Viewpoint (standard reference, not scraped)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)

Sphere self-maps of degree ±1±1 are exactly the homotopy equivalences