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Middle form and signature of the Milnor disk bundle

Statement

Assume the Axiom of Choice as inherited from the self-intersection and Thom suppliers. Let $W = W_{h, j} = D (ξ_{h, j})$ , $M = M_{h, j} = S (ξ_{h, j})$ and let $ε = h + j$ .

If $ε = \pm 1$ , then the boundary middle form of Boundary middle form and boundary signature on $W$ is the rank-one form $[ε]$ generated by the zero section, and the boundary signature is $σ (W) = ε$ .
For arbitrary $h, j$ the zero section of the disk bundle has self-intersection $h + j$ in the boundaryless interior; if $h + j = 0$ the rank-one homological zero-section form is degenerate, while the cohomological image $I$ is the zero space and its form has signature zero.

Facts & Assumptions

Given: The bundle $ξ_{h, j}$ over $S^{4}$ , the disk bundle $W = D (ξ_{h, j})$ with projection $π$ , zero section $s$ , sphere bundle $M$ , the classes $x = π^{*} u$ and the Thom generator $U$ .

[A1]

The Axiom of Choice is assumed (The Axiom of Choice).

[L1]

The Thom isomorphism gives $H^{4} (W, M; Z) = Z \cdot U$ and $H^{4} (W; Z) = Z \cdot x$ , and $j (U) = e (ξ_{h, j}) \cdot (generator) = (h + j) x$ in the calibrated conventions (Thom isomorphism for oriented vector bundles, Euler and first Pontryagin classes of $ξ_{h, j}$ , The Milnor sphere and disk bundles $M_{h, j}$ and $W_{h, j}$ ).

[L2]

When $ε = \pm 1$ , the normalized Thom evaluation satisfies $⟨ U ⌣ x, [W, M] ⟩ = 1$ (The Thom class of a disk bundle pairs with the base generator to one).

[L3]

The boundary middle form is $Q_{W} (y, y^{'}) = ⟨ \tilde{y} ⌣ {\tilde{y}}^{'}, [W, M] ⟩$ on $I = im j$ , with signature defined by inertia (Boundary middle form and boundary signature).

[L4]

Assume AC. For a compact boundaryless oriented embedded $a$ -submanifold in an oriented boundaryless $2 a$ -manifold, with the induced orientation on its normal bundle $ν$ , the self-intersection number equals the evaluation of the Euler class of $ν$ (The self-intersection number is the Euler number of the normal bundle).

Proof

technique · direct

1.1L1A1

By [L1] the map $j$ is multiplication by $ε = h + j$ on the infinite cyclic group generated by $U$ ; hence $I = im j$ over $R$ is $R \cdot x$ when $ε = \pm 1$ and $I = 0$ when $ε = 0$ , and the relative lift of $x$ is $ε U$ in the first case.

2.1step 1.1L2L3

If $ε = \pm 1$ , then $Q_{W} (x, x) = ⟨ (ε U) ⌣ x, [W, M] ⟩ = ε ⟨ U ⌣ x, [W, M] ⟩ = ε$ by [L2] and [L3]; the form is therefore rank one on $I = R \cdot x$ with matrix $[ε]$ , and its inertia signature is $ε$ .

3.1step 2.1L3

If $ε = 0$ , then $j = 0$ by step 1.1, so $I = 0$ ; the induced form on the zero-dimensional space $I$ is nondegenerate with no positive or negative directions, so its signature is zero, while the homological rank-one zero-section form has matrix $[0]$ and is degenerate. The radical of this zero-space form is zero; the radical of the homological rank-one form is its entire one-dimensional space.

4.1step 3.1L1L4

For arbitrary $h, j$ the zero section $s : S^{4} \to W$ is a closed oriented embedded submanifold of the boundaryless interior with normal bundle $ξ_{h, j}$ , so by [L4] its self-intersection number is $⟨ e (ξ_{h, j}), [S^{4}] ⟩ = h + j$ .

5.1step 4.1∎

Therefore for $ε = \pm 1$ the middle form is $[ε]$ with signature $ε$ , and for $h + j = 0$ the cohomological image form is zero-dimensional with signature zero while the homological zero-section form is degenerate, as asserted.

Depends on

Used by

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Sources

John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics 64 (1956), 399-405 (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Cambridge University Press 2002 (complete book) (standard reference, not scraped)