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Characteristic numbers of products satisfy the Whitney-sum and Kunneth product formulas

Statement

Assume AC (The Axiom of Choice), inherited from the Kunneth, Whitney-sum, Pontryagin-multiplicativity and characteristic-number suppliers, and used only there. Let $M^{m}$ and $N^{n}$ be closed smooth manifolds, and write $T (M \times N) ≅ T M ⊞ T N$ for the canonical splitting of the tangent bundle of a product (Canonical tangent and cotangent splittings for products). The total Stiefel-Whitney class is multiplicative under the Kunneth cross product, $w (T (M \times N)) = w (T M) \times w (T N) .$ For the Pontryagin classes the identity $p (T (M \times N)) = p (T M) \times p (T N)$ holds over $Z [1 / 2]$ , and it holds integrally whenever the odd Chern classes of $T M_{C}$ and $T N_{C}$ vanish, in particular when $M$ and $N$ are complex manifolds; integrally the difference is the two-torsion cross term $p (T (M \times N)) - p (T M) \times p (T N) = \sum_{a, b \geq 0} (- 1)^{a + b + 1} c_{2 a + 1} (T M_{C}) \times c_{2 b + 1} (T N_{C}) .$ In all cases the characteristic numbers expand by splitting each labeled index. For $I = (i_{1}, \dots, i_{t})$ with $\sum_{l} i_{l} = m + n$ , $w^{I} [M \times N] = \sum_{a_{l} + b_{l} = i_{l}} w_{a_{1}} \dots w_{a_{t}} [M] w_{b_{1}} \dots w_{b_{t}} [N] \in F_{2} .$ For closed oriented $M^{4 a}, N^{4 b}$ and $J = (j_{1}, \dots, j_{t})$ with $\sum_{l} j_{l} = a + b$ , $p_{J} [M \times N] = \sum_{a_{l} + b_{l} = j_{l}} p_{a_{1}} \dots p_{a_{t}} [M] p_{b_{1}} \dots p_{b_{t}} [N] \in Z .$ Each sum runs over nonnegative pairs independently for every $l$ . Zero-index classes are $1$ and are omitted from the resulting partitions; a factor monomial of the wrong degree contributes zero. In particular indices may split nontrivially, such as $2 = 1 + 1$ . A cross product of specified top-degree monomials from the two factors evaluates to the product of their numbers.

Facts & Assumptions

Given: Closed smooth manifolds $M^{m}$ and $N^{n}$ and the product $M \times N$ with its product smooth structure; oriented structures where Pontryagin numbers occur, with $m = 4 a$ , $n = 4 b$ in that case.

[F1]

Canonical tangent and cotangent splittings for products gives the canonical isomorphism $T_{(p, q)} (M \times N) ≅ T_{p} M \oplus T_{q} N$ , hence a canonical bundle isomorphism $T (M \times N) ≅ π_{1}^{*} T M \oplus π_{2}^{*} T N$ over the projections.

[F2]

Stiefel-Whitney numbers of a closed manifold and Pontryagin numbers of a closed oriented manifold define the characteristic numbers as evaluations on the fundamental class, componentwise over components, with the conventions $w_{0} = 1$ , $w_{i} = 0$ for $i > \dim$ , $p_{0} = 1$ , $p_{i} = 0$ for $2 i > rank$ , and with the value $0$ assigned to monomials of the wrong total degree.

[F3]

Whitney sum formula for Stiefel–Whitney classes gives the mod-two Whitney formula $w (E \oplus F) = w (E) w (F)$ and the trivial-summand stability over the admissible bases of that theorem; Naturality of Stiefel–Whitney classes gives naturality under pullback and invariance under bundle isomorphism.

[F4]

Pontryagin classes by complexification defines $p_{i} (E) = (- 1)^{i} c_{2 i} (E_{C})$ ; Naturality, stability, and mod-two reduction of Pontryagin classes gives naturality, stability and the rank cutoff; Naturality, normalization, and Whitney sum for Chern classes gives naturality and the integral Whitney formula for Chern classes; Odd Chern classes of a complexified real bundle are two-torsion gives $2 c_{2 j + 1} (E_{C}) = 0$ ; Complexification is conjugation invariant gives $c_{i} (\overline{V}) = (- 1)^{i} c_{i} (V)$ ; Pontryagin Whitney product away from two gives $p (E \oplus F) = p (E) p (F)$ over $Z [1 / 2]$ and asserts no integral multiplicativity.

[F5]

The fundamental class of a product is the cross product of the fundamental classes gives $[M \times N] = [M] \times [N]$ for the product orientation, and over $F_{2}$ for the canonical mod-two orientations; The Kronecker pairing is multiplicative under cross products gives $⟨ α \times β, c \times d ⟩ = ⟨ α, c ⟩ ⟨ β, d ⟩$ .

[F6]

Kronecker evaluation pairing and The kronecker pairing is independent of cocycle and cycle representatives make the pairing well defined and biadditive, so a class $x$ with $2 x = 0$ pairs to zero with every integral homology class, since $2 ⟨ x, z ⟩ = ⟨ 2 x, z ⟩ = 0$ in $Z$ ; Cohomological Kunneth cross product is a ring isomorphism defines the external product $a \times b = {pr}_{1}^{*} a ⌣ {pr}_{2}^{*} b$ and its multiplication $(a \times b) (a^{'} \times b^{'}) = (- 1)^{∣ b ∣ ∣ a^{'} ∣} (a a^{'}) \times (b b^{'})$ ; Field Kunneth isomorphism for homology of products gives the field-coefficient Kunneth isomorphism used for the mod-two evaluations.

[F7]

Top homology of a connected manifold gives homology vanishing above the dimension on each compact connected component; Cohomology over a field is dual to homology over that field gives the corresponding mod-two cohomology vanishing under AC. The Pontryagin-number definition gives CW-type transport of naturality and stability; the same transport, using the Chern Whitney and conjugation identities on a CW model, gives [F4] on smooth-manifold bases.

Proof

1.1F1F2F3F5F6F7

By [F1, F3], $w (T (M \times N)) = π_{1}^{*} w (T M) π_{2}^{*} w (T N) = w (T M) \times w (T N)$ . Thus $w_{i_{l}} (T (M \times N)) = \sum_{a_{l} + b_{l} = i_{l}} w_{a_{l}} (T M) \times w_{b_{l}} (T N)$ . Multiplying these finite sums gives the displayed indexed expansion; Koszul signs disappear over $F_{2}$ . By [F5] each term whose factor degrees are $m, n$ evaluates to the product of its factor numbers. A factor class above its manifold dimension is zero by top-homology vanishing and field duality; since the two degrees sum to $m + n$ , every term with unequal factor degrees has such an over-dimension factor. Hence precisely the wrong-degree terms contribute zero, as stipulated in [F2].

2.1F1F4step 1.1

Pontryagin defect. Complexifying the splitting [F1] and using naturality and the Whitney formula for Chern classes [F4] gives $c ((T (M \times N))_{C}) = π_{1}^{*} c (T M_{C}) \cdot π_{2}^{*} c (T N_{C})$ . Writing $c (T M_{C}) = \sum_{i} c_{i}$ and $c (T N_{C}) = \sum_{j} c_{j}$ and comparing even parts with the definition $p_{k} = (- 1)^{k} c_{2 k}$ [F4] gives $p (T (M \times N)) - p (T M) \times p (T N) = \sum_{a, b \geq 0} (- 1)^{a + b + 1} c_{2 a + 1} (T M_{C}) \times c_{2 b + 1} (T N_{C}),$ because the even-even terms reassemble to the cross product of the two total Pontryagin classes and each odd-odd term appears with the sign recorded. Each odd Chern class of a complexified real bundle is two-torsion by [F4], so the right-hand side, a sum of cross products of two-torsion classes, is two-torsion; hence it vanishes in $H^{*} (M \times N; Z [1 / 2])$ , giving the stated identity over $Z [1 / 2]$ , which also follows directly from the away-from-two multiplicativity in [F4]. If the odd Chern classes of $T M_{C}$ and $T N_{C}$ all vanish the correction is zero, so the identity is integral.

3.1F4step 2.1

The complex-manifold case. If $M$ and $N$ are complex manifolds, then $T M$ and $T N$ are complex vector bundles, and the complexification of an underlying real complex bundle is $V \oplus \overline{V}$ : the map $v \otimes z \mapsto (z v, z \overset{ˉ}{v})$ is complex linear, with inverse $(a, \overset{ˉ}{b}) \mapsto \frac{a + b}{2} \otimes 1 + \frac{a - b}{2 i} \otimes i$ , where $v \mapsto \overset{ˉ}{v}$ is the canonical antilinear copy. The formulas respect local frames. Thus $(T M_{R})_{C} ≅ T M \oplus \overline{T M}$ ; by the conjugation formula of [F4], $c (\overline{T M})$ is obtained from $c (T M)$ by $c_{i} \mapsto (- 1)^{i} c_{i}$ , so the odd part of $c (T M \oplus \overline{T M})$ cancels in pairs and the correction of step 2.1 vanishes. Hence the integral class identity $p (T (M \times N)) = p (T M) \times p (T N)$ holds for complex-manifold factors, and in particular for products of complex projective spaces.

3.2F2F4F5F6F7step 2.1

For $J = (j_{1}, \dots, j_{t})$ , write $p_{j_{l}} (T (M \times N)) = \sum_{a_{l} + b_{l} = j_{l}} p_{a_{l}} (T M) \times p_{b_{l}} (T N) + δ_{j_{l}}$ , where each $δ_{j_{l}}$ is two-torsion by step 2.1. In the product every term containing a correction remains two-torsion and pairs to zero with the integral fundamental class by [F6]. The remaining product is the product of these individually prescribed homogeneous sums, and expands over all tuples $(a_{l}, b_{l})$ in the statement, with positive Koszul signs. Terms of bidegree $(4 a, 4 b)$ evaluate by [F5] to the product of the two factor numbers. For any other bidegree of the same total degree one factor exceeds its dimension; over $Q$ it vanishes by top-homology vanishing and field duality [F7], so its integral product term has zero integral evaluation by coefficient naturality and injectivity of $Z \to Q$ [F6]. This proves the integer formula, including the wrong-degree convention.

4.1F2F5step 1.1step 3.2∎

The evaluation of a specified cross product of top-degree monomials is the single product of evaluations by [F5]. When one factor is zero-dimensional, the formulas reduce componentwise to the sum of signed point contributions over $Z$ or point parity over $F_{2}$ ; these need not equal $1$ . Empty factors give zero. No choice beyond the stated supplier assumptions is used.

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Sources

John Milnor and James Stasheff, Characteristic Classes (original pagination; chapters 16-18 of the re-typeset scan) (standard reference, not scraped)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)