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The mod 2 intersection number is homotopy invariant

Statement

Assume ${A C}_{ω}$ . Let $X$ be a compact smooth manifold without boundary, $M$ a smooth $n$ -manifold, $Z \subseteq M$ a closed embedded submanifold, and $F : [0, 1] \times X \to M$ a smooth family with $\dim X + \dim Z = n$ . Assume $F$ is transverse to $Z$ , including on the boundary faces ${0} \times X$ and ${1} \times X$ in the sense of Transverse preimages for maps from manifolds with boundary. Then the slice maps $F_{0} = F (0, \cdot)$ and $F_{1} = F (1, \cdot)$ are transverse to $Z$ , $F^{- 1} (Z)$ is a compact $1$ -manifold with boundary $\partial F^{- 1} (Z) = F_{0}^{- 1} (Z) ⊔ F_{1}^{- 1} (Z),$ and $# F_{0}^{- 1} (Z) \equiv # F_{1}^{- 1} (Z) (m o d 2), that is, I_{2} (F_{0}, Z) = I_{2} (F_{1}, Z) .$ Consequently $I_{2}$ of arbitrary smooth maps is well defined and homotopy invariant, and for transverse representatives the geometric parity of the intersection is the invariant.

Facts & Assumptions

Given: A compact boundaryless $X$ , a closed embedded $Z \subseteq M$ with $\dim X + \dim Z = \dim M$ , and a smooth family $F : [0, 1] \times X \to M$ transverse to $Z$ including on the faces.

[F1]

$[0, 1] \times X$ is a smooth manifold with boundary ${0} \times X ⊔ {1} \times X$ , and $F$ is a smooth family in the sense of the evaluation map (Products of smooth manifolds have a canonical product smooth structure, Smooth families of maps and their evaluation maps).

[F2]

Under these hypotheses $F^{- 1} (Z)$ is an embedded submanifold with boundary of $[0, 1] \times X$ , neat, of dimension $\dim X + 1 + \dim Z - \dim M = 1$ , with boundary $F_{0}^{- 1} (Z) ⊔ F_{1}^{- 1} (Z)$ and $T_{p} F^{- 1} (Z) = {v : d F_{p} (v) \in T_{F (p)} Z}$ (Transverse preimages for maps from manifolds with boundary).

[F3]

The slice preimages $F_{0}^{- 1} (Z)$ and $F_{1}^{- 1} (Z)$ are finite, and $I_{2} (f, Z) = # f^{- 1} (Z) m o d 2$ for a transverse $f$ (Compact transverse complementary intersections are finite, The mod 2 intersection number).

[F4]

The boundary of a compact $1$ -manifold has even cardinality (Boundary of a compact 1-manifold has even cardinality).

[F5]

Congruence modulo $2$ is additive: equalities of integers may be reduced termwise (The congruence class $[a]_{n}$ and the quotient set $Z / n$ ).

[F6]

Under ${A C}_{ω}$ , The transversality homotopy theorem supplies transverse representatives and Relative Whitney approximation for manifold-valued maps smooths a continuous homotopy fixed near its ends. For a smooth map on the boundaryless extension $R \times X$ , A tubular target produces a submersive finite-dimensional perturbation family supplies a parameter ball centred at $0$ whose parameter maps are submersions; Parametric transversality makes the bad-slice parameters a null set. A positive-dimensional ball is not null; in dimension zero a null subset is empty. These suppliers and the classification in [F4] assume The Axiom of Countable Choice ( ${A C}_{ω}$ ).

Proof

technique · direct, by counting the boundary of the compact trace

1.1F1F2F3given

By [F1] and [F2] the trace $W : = F^{- 1} (Z)$ is a compact embedded submanifold with boundary of $[0, 1] \times X$ of dimension $1$ , with $\partial W = F_{0}^{- 1} (Z) ⊔ F_{1}^{- 1} (Z)$ ; compactness follows because $W$ is closed in the compact product $[0, 1] \times X$ . In particular the two slice preimages are finite by [F3].

2.1F3F4F5step 1.1algebra

By [F4] the boundary of the compact $1$ -manifold $W$ has even cardinality, so $# \partial W = # F_{0}^{- 1} (Z) + # F_{1}^{- 1} (Z)$ is even. Reducing modulo two and using [F5] gives $# F_{0}^{- 1} (Z) \equiv # F_{1}^{- 1} (Z) (m o d 2)$ , that is, $I_{2} (F_{0}, Z) = I_{2} (F_{1}, Z)$ by [F3].

3.1F3F4F6step 2.1constructchoosealgebra∎

Given a continuous homotopy between transverse endpoints, reparametrize by a smooth function constant in end collars, extend constantly to $R \times X$ , and smooth it fixed on smaller closed end regions by [F6]. Call the smooth result $H$ ; it is constant at its transverse endpoints for $t \leq δ$ and $t \geq 1 - δ$ , for some $0 < δ < 1 / 2$ . Take the submersive parameter family $H$ for $H$ from [F6], with centred ball $B$ . Choose a smooth $λ : R \to [0, 1]$ equal to $0$ outside $(δ / 2, 1 - δ / 2)$ and positive inside, and set $G ((t, x), a) = H ((t, x), λ (t)^{2} a)$ . Where $λ > 0$ the parameter derivative is surjective; where $λ = 0$ , $d (λ^{2}) = 0$ and the source derivative is that of the already transverse endpoint map. Thus $G$ is transverse to $Z$ . By parametric transversality choose a good $a \in B$ (also when $B$ is zero-dimensional); its slice, restricted to $[0, 1] \times X$ , is transverse and has the original endpoints. Applying 2.1 to this fixed-endpoint transverse homotopy proves equality of the parities of any homotopic transverse representatives. Representatives exist by [F6], so $I_{2}$ is well defined and homotopy invariant. For two compact-source maps, product homotopies and the diagonal definition give invariance under deformation of either or both factors. Countable Choice is inherited from the classification and approximation suppliers; the finite parity computation adds no choice.

Depends on

Used by

The oriented intersection number reduces to the mod 2 number Corollary
Geometric cardinality is not homotopy invariant Counterexample
Noncompact intersections can escape during a homotopy Counterexample
Latitude and meridian intersections on the torus Example
Two projective lines have one mod 2 intersection Example
Properness can replace compactness only when the intersection trace is compact Remark
The oriented intersection number is homotopy invariant Theorem

Cited to discharge well-definedness by The mod 2 intersection number.

Dependency tree · two levels

71 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 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)