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A sphere reflection has degree minus one

Example

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For $m \geq 1$ let $R (x_{0}, x_{1}, \dots, x_{m}) = (- x_{0}, x_{1}, \dots, x_{m})$ be the coordinate reflection of $S^{m}$ , oriented by the outward-normal-first boundary orientation of $S^{m} = \partial D^{m + 1}$ . Then $R$ is a smooth diffeomorphism of $S^{m}$ reversing orientation, so $\deg (R) = - 1$ ; by the sphere classification every self-map of $S^{m}$ of degree $- 1$ is homotopic to $R$ , and by the homotopy-equivalence corollary a self-map of $S^{m}$ of degree $- 1$ is a homotopy equivalence exactly when it is homotopic to a reflection.

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 $m \geq 1$ , the sphere $S^{m} = \partial D^{m + 1}$ with its outward-normal-first orientation and the coordinate reflection $R$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ , the local calculation, Induced boundary orientation).

[F1]

The coordinate reflection restricts to an orientation-reversing smooth diffeomorphism of $S^{m}$ and has degree $- 1$ (the local calculation, the local calculation).

[F2]

An orientation-reversing diffeomorphism between nonempty connected oriented boundaryless manifolds has degree $- 1$ (Degree of an orientation-preserving or reversing diffeomorphism).

[F3]

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).

[F4]

A self-map of $S^{m}$ is a homotopy equivalence exactly when its degree is $\pm 1$ (Sphere self-maps of degree $\pm 1$ are exactly the homotopy equivalences).

Verification

technique · direct

1.1L1F1F2given

The reflection $R$ is the restriction of the invertible linear map of $R^{m + 1}$ with determinant $- 1$ that preserves the unit sphere, hence restricts to a smooth diffeomorphism of $S^{m}$ ; the linear reflection reverses the ambient orientation and the outward-normal-first orientation of $S^{m}$ is transported from the ambient orientation, so $R$ reverses the orientation of $S^{m}$ , and $\deg (R) = - 1$ both by the direct reflection computation of [F1] and by the diffeomorphism criterion of [F2].

2.1F3F4step 1.1

Let $f : S^{m} \to S^{m}$ have degree $- 1$ . By [F3] applied to $f$ and $R$ , whose degrees are equal, $f$ is homotopic to $R$ ; conversely every map homotopic to $R$ has degree $- 1$ by [F3]. By [F4] every self-map of degree $- 1$ is a homotopy equivalence, and by [F3] its homotopy class is that of the reflection, so a map of degree $- 1$ is a homotopy equivalence precisely when it lies in the homotopy class of a reflection.

3.1F1F3step 2.1∎

This identifies the degree- $- 1$ class, represented by a reflection, and shows the consistency of the reflection sign with the general classification; no new invariant or orientation convention is introduced beyond the outward-normal-first orientation of the sphere.

Depends on

Used by

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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)