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The relative Pontryagin square glues across a seven-dimensional boundary

Statement

Assume the Axiom of Choice as inherited from the duality and characteristic-class suppliers. Let $M$ be a closed oriented smooth seven-manifold with $H^{3} (M; Z) = H^{4} (M; Z) = 0$ , let $W, W^{'}$ be compact oriented smooth eight-manifolds with $\partial W = M = \partial W^{'}$ , and put $N = W \cup_{M} (- W^{'})$ . Then $⟨ p_{1} (T N)^{2}, [N] ⟩ = q (W) - q (W^{'}),$ where $q$ is the relative square of The Milnor lambda candidate from a supplied filling. Orientation reversal changes the evaluations on the two sides, not the Pontryagin class itself.

Facts & Assumptions

Given: The manifold $M$ with $H^{3} (M; Z) = H^{4} (M; Z) = 0$ , the two fillings $W, W^{'}$ and the closed oriented glued manifold $N = W \cup_{M} (- W^{'})$ .

[A1]

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

[L1]

Under the vanishing hypotheses the relative-to-absolute maps $j_{W} : H^{4} (W, M; Z) \to H^{4} (W; Z)$ and $j_{W^{'}}$ are isomorphisms; the classes ${\overset{ˉ}{p}}_{1} (W) = j_{W}^{- 1} p_{1} (T W)$ and $q (W) = ⟨ {\overset{ˉ}{p}}_{1} (W) ⌣ {\overset{ˉ}{p}}_{1} (W), [W, M] ⟩$ are defined by The Milnor lambda candidate from a supplied filling, and likewise on the second side. [A1, given]

[L2]

For the collared gluing $N = W \cup_{M} (- W^{'})$ with collar cover $U, V$ there are excision isomorphisms and evaluation comparisons, and $[N]$ restricts to $[W, M]$ on the first side and to $- [W^{'}, M]$ on the second (Collared gluing has relative excision and evaluation maps).

[L3]

Mayer-Vietoris for the open cover $U, V$ gives the exact segment $H^{3} (U \cap V) \to H^{4} (N) \to H^{4} (U) \oplus H^{4} (V) \to H^{4} (U \cap V)$ , and the pair sequences give the exactness used in [L1] (Mayer vietoris sequence in singular cohomology, Long exact sequence of a pair in singular cohomology).

[L4]

Pontryagin classes are natural on CW bases and $p_{1}$ is orientation-independent (Naturality, stability, and mod-two reduction of Pontryagin classes). AC implies ${A C}_{ω}$ , so all compact smooth manifolds here have finite CW homotopy models (AC implies DC implies countable choice, Compact smooth manifolds have finite CW models under countable choice). Classes on path-connected CW-type bases are defined by transport (Pontryagin classes by complexification). For a map $f : A \to B$ , choose model equivalences $a : X \to A$ , $b : Y \to B$ and a homotopy inverse $b^{'}$ of $b$ . Then $b b^{'} f a ≃ f a$ , so their pulled-back bundles are isomorphic (Homotopy invariance of vector-bundle pullback). Naturality for $b^{'} f a : X \to Y$ and invertibility of $a^{*}$ prove naturality for $f$ after transport. For a possibly disconnected compact smooth manifold, its finitely many open path-connected components each have finite CW type; define $p_{1}$ componentwise as in [L1]. The naturality calculation applies on each component, and the singular-cochain product decomposition assembles the resulting equalities. Thus the same formula applies to the inclusions of the smooth halves here, including disconnected fillings.

[L5]

Relative cup products are natural and compatible with the excision and connector maps (Relative cup product for an excisive triad, Relative cup products are natural and connector-compatible).

Proof

technique · direct

1.1L1A1

By [L1] the maps $j_{W}, j_{W^{'}}$ are integral isomorphisms, so ${\overset{ˉ}{p}}_{1} (W)$ and ${\overset{ˉ}{p}}_{1} (W^{'})$ and the relative squares $q (W), q (W^{'})$ are defined.

2.1step 1.1L2L3

With the collar cover $U, V$ of [L2], the identifications $U ≃ W$ , $V ≃ W^{'}$ , $U \cap V ≃ M$ turn the Mayer-Vietoris segment [L3] into $0 = H^{3} (M) \to H^{4} (N) \to H^{4} (W) \oplus H^{4} (W^{'}) \to H^{4} (M) = 0$ , so restriction to the two halves is an isomorphism $H^{4} (N) ≅ H^{4} (W) \oplus H^{4} (W^{'})$ .

3.1step 2.1L2

For $x \in H^{4} (W)$ define $L_{W} (x) = j_{V} E_{W} (j_{W}^{- 1} x)$ , using the inverse excision map of [L2], and similarly $L_{W^{'}} (x^{'}) = j_{U} E_{W^{'}} (j_{W^{'}}^{- 1} x^{'})$ ; their restrictions are $(x, 0)$ and $(0, x^{'})$ respectively, because the extended relative class vanishes on the opposite side, and by step 2.1 they are the unique classes with those restrictions.

3.2step 2.1L2L4

The tangent bundles of $N$ restrict to $T W$ and $T W^{'}$ on the two halves: on a collar both tangent bundles are $T M \oplus R$ with the normal coordinate reversed at the seam, and the derivative of the gluing identification gives a real bundle isomorphism; by [L4] the Pontryagin class is unchanged, so $p_{1} (T N)$ restricts to $p_{1} (T W)$ and $p_{1} (T W^{'})$ .

4.1step 3.1step 3.2

By step 3.1 and step 3.2 the class $L_{W} (p_{1} (T W)) + L_{W^{'}} (p_{1} (T W^{'}))$ has the same restrictions as $p_{1} (T N)$ ; the uniqueness in step 2.1 gives $p_{1} (T N) = L_{W} (p_{1} (T W)) + L_{W^{'}} (p_{1} (T W^{'}))$ .

4.2step 3.1L2L5

For relative classes $a \in H^{4} (W, M; Z)$ and $b \in H^{4} (W^{'}, M; Z)$ , the product $E_{W} (a) ⌣ E_{W^{'}} (b) \in H^{8} (N, V \cup U) = H^{8} (N, N) = 0$ lies in a zero group, because $U \cup V = N$ ; hence the cross products $L_{W} (x) ⌣ L_{W^{'}} (x^{'})$ and its reverse vanish in $H^{8} (N; Z)$ .

5.1step 4.1step 4.2L2L5

By the evaluation comparison of [L2] and naturality of the relative products [L5], the same-side evaluation satisfies $⟨ L_{W} (x) ⌣ L_{W} (y), [N] ⟩ = ⟨ j_{W}^{- 1} x ⌣ y, [W, M] ⟩$ , which for $x = y = p_{1} (T W)$ is $q (W)$ ; on the second side the restriction of $[N]$ is $- [W^{'}, M]$ , so the evaluation is $- q (W^{'})$ .

6.1step 4.1step 4.2step 5.1∎

Expanding the square in step 4.1 and using that both cross products vanish by step 4.2 gives $p_{1} (T N)^{2} = L_{W} (p_{1} (T W))^{2} + L_{W^{'}} (p_{1} (T W^{'}))^{2}$ ; evaluating on $[N]$ with step 5.1 yields $⟨ p_{1} (T N)^{2}, [N] ⟩ = q (W) - q (W^{'})$ , as asserted.

Depends on

Used by

The Milnor lambda invariant is well defined modulo seven Theorem

Dependency tree · two levels

68 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 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)