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Preimage orientation agrees with the local intersection sign

Statement

Let $f : Y \to M$ be transverse to a closed oriented embedded submanifold $Z$ , with $Y, M$ oriented, $M$ boundaryless, and, if $Y$ has boundary, $f ∣_{\partial Y}$ transverse to $Z$ . Orient $Q_{y} = T_{f (y)} M / T_{f (y)} Z$ by the normal-first determinant isomorphism $\det Q_{y} \otimes \det T_{f (y)} Z ≅ \det T_{f (y)} M$ : quotient lifts precede the tangent determinant of $Z$ . Orient $K_{y} = T_{y} f^{- 1} (Z)$ by the kernel-first exact-sequence convention $\det T_{y} Y ≅ \det K_{y} \otimes \det Q_{y},$ where $d f$ induces the quotient map. In complementary dimensions $\dim Y + \dim Z = \dim M$ , the resulting point sign is the local oriented intersection sign. If $Z = {z}$ has positive point orientation, this sign is $sgn (d f_{y})$ ; reversing that point orientation reverses the intersection sign.

Facts & Assumptions

Given: Oriented $Y, M, Z$ , a transverse $f : Y \to M$ , and $y \in f^{- 1} (Z)$ .

[F1]

The transverse preimage tangent is $K_{y} = {v : d f_{y} (v) \in T_{f (y)} Z}$ ; thus $0 \to K_{y} \to T_{y} Y \to Q_{y} \to 0$ is exact (The transverse preimage theorem, and Transverse preimages for maps from manifolds with boundary for a boundary source).

[F2]

Orientation rays and ordered product determinants are the conventions of Determinant-line orientations of finite-dimensional real vector spaces and Product orientations.

[F3]

In complementary dimensions the local sign compares $(d f_{y}, d i) : T_{y} Y \oplus T_{f (y)} Z \to T_{f (y)} M$ with its given orientations (The local oriented intersection sign).

[F4]

The local sign of an equidimensional regular preimage compares the supplied source and target determinant rays, including point signs in dimension zero (Local orientation sign of a regular preimage).

Proof

1.1F1F2givenconstruct

For the normal-first isomorphism, wedge lifts of a quotient determinant before a tangent determinant of $Z$ . Replacing a lift by a tangent vector changes the wedge by zero, so this is independent of lifts. Similarly wedging a kernel determinant before lifts of a quotient determinant defines the kernel-first exact-sequence isomorphism in the statement; it is independent of lifts and smooth in local adapted frames. The given rays therefore determine a unique smooth orientation of $K$ .

2.1F1F2F3step 1.1algebra

In complementary dimensions $K_{y} = 0$ . Take positive determinant elements $u$ of $T_{y} Y$ and $w$ of $T_{f (y)} Z$ . The sign $ε$ of $(d f_{y} u) \land w$ relative to the ambient ray is exactly the local intersection sign. By the normal-first convention, ${\overline{d f}}_{y} (u)$ has sign $ε$ relative to the quotient ray. In $\det T_{y} Y = \det K_{y} \otimes \det Q_{y}$ , the chosen scalar ray of $\det K_{y} = R$ must therefore have sign $ε$ too, so that the product ray is the given source ray. This is precisely the point orientation of the fibre.

3.1F2F4step 2.1algebra∎

For a positively oriented point $Z$ , the normal-first quotient ray is the ambient ray, so 2.1 gives $sgn (d f_{y})$ by [F4]. A negatively oriented point changes that quotient ray and hence the fibre sign. All computations use determinant elements rather than positive empty bases and therefore include dimension zero; no choice principle is needed.

Depends on

Used by

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Sources

Victor Guillemin and Alan Pollack, Differential Topology (Prentice-Hall, 1974; complete 236-page PDF) (standard reference, not scraped)
John Milnor, Topology from the Differentiable Viewpoint (Princeton University Press; complete 76-page PDF, including the appendix Classifying 1-manifolds) (standard reference, not scraped)