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The index of a zero is its zero-section intersection number

Statement

Assume countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $M$ be a closed oriented smooth $n$ -manifold, $n \geq 1$ , and give $T M$ the orientation which along the zero section is the direct sum of the horizontal (tangent) orientation of the base and the vertical (fibre) orientation, first factor first (Product orientations, The tangent bundle as a disjoint union), orient both submanifolds by their projections to $M$ , and let $X_{0} \subset T M$ be the zero section (The zero section is a smooth embedding) and let $Γ_{X}$ be the graph of a smooth vector field $X$ transverse to $X_{0}$ , so that the zeros of $X$ are nondegenerate. Then, with the oriented intersection number of The oriented intersection number, $I (X_{0}, Γ_{X}) = \sum_{p : X (p) = 0} {ind}_{p} X,$ and at each zero the local oriented intersection sign of The local oriented intersection sign satisfies $ε_{(X_{0}, Γ_{X})} (p) = {ind}_{p} X$ .

Facts & Assumptions

Given: A closed oriented smooth $n$ -manifold $M$ , the tangent bundle with its horizontal-then-vertical orientation along the zero section, and a smooth field $X$ whose graph is transverse to the zero section.

[F1]

$X_{0}$ and $Γ_{X}$ are closed embedded $n$ -submanifolds of the boundaryless $2 n$ -manifold $T M$ with complementary dimensions, and for a transverse pair with one factor compact the intersection is finite (The zero section is a smooth embedding, Transverse embedded submanifolds, Compact transverse complementary intersections are finite).

[F2]

For a compact oriented $x$ -manifold without boundary $A$ , an oriented $n$ -manifold $M$ and a closed oriented embedded $z$ -submanifold $Z$ with $x + z = n$ and transverse inclusion, the oriented intersection number is the finite sum $I (A, Z) = \sum_{p \in A \cap Z} ε (p)$ of the local signs of The local oriented intersection sign, with the product orientation on the ordered sum of tangent spaces, first factor first (The oriented intersection number).

[F3]

Local model of the intersection: over a chart $U$ of $M$ in which $T M$ is trivialized as $U \times R^{n}$ , the zero section is $U \times {0}$ and the graph is ${(u, X_{φ} (u))}$ ; at a point $p \in X^{- 1} (0)$ the tangent spaces are $T_{p} M \oplus {0}$ and ${(v, D X_{p} v)}$ , and the determinant comparing the ordered product $T_{p} X_{0} \oplus T_{(p, 0)} Γ_{X}$ with the ambient horizontal-then-vertical orientation is $\det (D X_{p})$ : the relevant matrix is block-triangular with identity and $D X_{p}$ blocks, exactly as in the push-off computation of Normal push-off zeros are the self-intersection points (The induced tangent bundle chart, Nondegenerate zero of a vector field).

[F4]

The zeros of $X$ are exactly the intersection points of $Γ_{X}$ with $X_{0}$ , and $Γ_{X}$ is transverse to $X_{0}$ exactly when $D X_{p}$ is invertible at every zero, equivalently when every zero is nondegenerate; a nondegenerate zero has index $sign \det (D X_{p}) = \pm 1$ (Nondegenerate zero of a vector field, The index of a nondegenerate vector-field zero).

Proof

1.1F1F3F4algebra

Write $Γ_{X} = {(x, X (x)) : x \in M} \subseteq T M$ . Since $Γ_{X} \cap X_{0} = {(p, 0) : X (p) = 0}$ , the intersection points are the zeros of $X$ ; at such a point the transversality of $Γ_{X}$ to $X_{0}$ is equivalent to the surjectivity of $D X_{p}$ (the tangent spaces are the horizontal space and the graph of $D X_{p}$ ), so transversality means invertibility of $D X_{p}$ at every zero, i.e. nondegeneracy; by [F1] the intersection is finite, and by [F4] each intersection point carries index $sign \det (D X_{p})$ .

2.1F2F3F4step 1.1algebra∎

At a zero $p$ , the local sign $ε_{(X_{0}, Γ_{X})} (p)$ is computed in the chart of [F3] as the determinant sign of the block-triangular matrix with diagonal blocks $I_{n}$ and $D X_{p}$ , hence equals $sign \det (D X_{p}) = {ind}_{p} X$ ; this proves the pointwise identity and, summing over the finitely many intersection points with [F2], also $I (X_{0}, Γ_{X}) = \sum_{p} {ind}_{p} X$ . The countable choice hypothesis is inherited from the transverse-representative selection in the definition of the oriented intersection number for non-transverse maps (The oriented intersection number, The oriented intersection number is homotopy invariant), although for the transverse pair considered here the sum is choice-free.

Depends on

Used by

The Euler number of the tangent bundle is the Euler characteristic Corollary

Dependency tree · two levels

67 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 (Prentice-Hall 1974; complete PDF) (standard reference, not scraped)
John W. Milnor and James D. Stasheff, Characteristic Classes (Princeton University Press, 1974; complete PDF) (standard reference, not scraped)