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The reflection of an outward field extends over the double

Statement

Assume countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $M$ be a compact smooth $n$ -manifold with boundary, $n \geq 1$ , and let $X$ be a smooth vector field that is nonzero and points strictly outward along $\partial M$ (Inward, outward, and boundary-tangent vectors). Choose the collar generated by the inward field $- X$ . Let $D M$ be the smooth double defined by this collar, with seam involution $τ$ interchanging the two labelled halves (The double of a smooth manifold with boundary, The double has a well-defined smooth structure). Then $X^{+} (y) = X (y) (y \in M_{+}), X^{+} (τ y) = - d τ_{y} (X (y)) (y \in M_{+})$ defines a smooth vector field on this double (an arbitrary fixed collar need not give a smooth field); its zeros are exactly the two copies of the zeros of $X$ , and for every isolated zero $p \in int M$ ${ind}_{τ p} (X^{+}) = (- 1)^{n} {ind}_{p} X .$

Facts & Assumptions

Given: A compact smooth $n$ -manifold $M$ with boundary, a smooth field $X$ nonzero and strictly outward along $\partial M$ , and the labelled double $(D M, τ)$ .

[F1]

$D M$ is the quotient of $M_{+} ⊔ M_{-}$ identifying the two copies of $\partial M$ , with $τ$ the involution interchanging the labelled halves; smooth collar data near the seam give it the structure of a smooth boundaryless manifold, and two collar choices give structures related by a diffeomorphism fixing the seam pointwise and preserving the halves (The double of a smooth manifold with boundary, The double has a well-defined smooth structure).

[F3]

Because $X$ is strictly outward and nonzero on the compact boundary, there is $δ > 0$ such that $X$ has no zero in the $δ$ -neighbourhood of $\partial M$ ; the zeros of $X$ therefore lie in the interior at positive distance from $\partial M$ (Inward, outward, and boundary-tangent vectors).

[F4]

The inward field $- X$ has smooth local forward semiflows at boundary points, using smooth coordinate extensions across the face (Inward-pointing fields have local forward semiflows at the boundary). Their differentials at time zero are invertible because $- X$ is transverse to the boundary. Compactness supplies a uniform short time. Uniqueness and strict inwardness make the map $c (x, u) = Φ_{u}^{- X} (x)$ injective: a trajectory cannot return to the boundary, since its boundary defining coordinate has positive derivative at any putative return. Thus $c$ is a global collar for short time, and $X = - \partial_{u}$ in its coordinates.

[F5]

The index is independent of the chart and of a trivialization with matching base and fibre orientations, and negation scales it by $(- 1)^{n}$ (The local index is independent of chart, ball and trivialization, Negation scales the local index by $(- 1)^{n}$ ).

Proof

1.1F1F4givenconstruct

If $\partial M = \emptyset$ , the double is the disjoint union of two copies, and the two fields are $X$ and $- X$ . Otherwise use the single global flow collar $c (x, u)$ of [F4] to define the double's smooth structure. Its seam charts have signed coordinate $u$ , with $τ (x, u) = (x, - u)$ . This is the collar-defined double of [F1], rather than a replacement of an already fixed smooth structure while keeping $X$ unchanged.

2.1F1F4step 1.1algebra

In these charts $X = - \partial_{u}$ on the first half. Reflection followed by negation gives $- d τ (- \partial_{u}) = - \partial_{u}$ on the second half. The prescriptions therefore agree at the seam and give one smooth nonzero expression there. Off the seam they are $X$ and the push-forward of $- X$ , so they are smooth globally.

3.1F3F5step 1.1step 2.1algebra∎

There are no seam zeros, and off the seam the zero set consists precisely of the two copies of $X^{- 1} (0)$ . For any isolated zero $p$ , a chart at $p$ transported by $τ$ identifies the second field with $- X$ . Chart invariance and [F5] give ${ind}_{τ p} X^{+} = {ind}_{p} (- X) = (- 1)^{n} {ind}_{p} X$ . The same computation applies to the two disjoint copies when the boundary is empty.

Depends on

Used by

The index sum of an outward field on an even-dimensional manifold Lemma

Dependency tree · two levels

48 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

John W. Milnor, Topology from the Differentiable Viewpoint (complete 76-page PDF, including the appendix Classifying 1-manifolds) (standard reference, not scraped)
Joel W. Robbin and Dietmar A. Salamon, Introduction to Differential Topology (web draft 2018, complete PDF) (standard reference, not scraped)