03.06.10 · modern-geometry / characteristic-classes

Stiefel-Whitney and Pontryagin numbers

shipped3 tiersLean: partial

Anchor (Master): Milnor-Stasheff §4 + §16 + §17 + Appendix A; Thom 1954 Comment. Math. Helv. 28; Hirzebruch 1956 Topological Methods §1.5

Intuition [Beginner]

Characteristic classes live in the cohomology of a manifold — they are bookkeeping for how its tangent bundle twists. But cohomology classes are abstract; you cannot tell two manifolds apart just by writing down "the second Stiefel-Whitney class of $M$ is some element of $H^{2}$ ." You need to extract numbers.

The trick is that a closed $n$ -manifold $M$ has a fundamental class — a top-degree homology element that integrates cohomology classes over $M$ . Pair a top-degree characteristic monomial with this fundamental class and you get a single number: an element of $Z /2$ for Stiefel-Whitney, an integer for Pontryagin. These numbers are called the Stiefel-Whitney numbers and the Pontryagin numbers of $M$ .

The miracle is that these numbers depend only on the bordism class of $M$ : two manifolds that together bound a one-dimensional-higher manifold share the same Stiefel-Whitney numbers (Thom 1954) and the same Pontryagin numbers (after orientation). The numerical invariants are a complete bordism fingerprint.

Visual [Beginner]

Think of a closed surface or 4-manifold $M$ . Its tangent bundle carries a small list of characteristic classes — $w_{1}, w_{2}, \dots$ in $Z /2$ -cohomology, plus $p_{1}, p_{2}, \dots$ in integral cohomology if $M$ is oriented and has dimension divisible by $4$ . The picture shows the top-degree monomials being evaluated against the fundamental class, returning a row of numbers.

The picture is the cohomological side of pairing. Bordism is the homological side: two manifolds with the same row of numbers are bordant.

Worked example [Beginner]

The complex projective plane $CP^{2}$ is a closed oriented 4-manifold. It has one Pontryagin class $p_{1} \in H^{4} (CP^{2}; Z)$ .

Step 1. The cohomology of $CP^{2}$ is $H^{*} (CP^{2}; Z) = Z [x] / (x^{3})$ with $x$ in degree $2$ .

Step 2. The Euler sequence gives $c (T CP^{2}) = (1 + x)^{3} = 1 + 3 x + 3 x^{2}$ , so $p_{1} (T CP^{2}) = 3 x^{2}$ after the standard Pontryagin-from-complexification calculation.

Step 3. The fundamental class pairs $x^{2}$ with $1$ , so the Pontryagin number is $p_{1} [CP^{2}] = ⟨ 3 x^{2}, [CP^{2}]⟩ = 3$ .

What this tells us: $CP^{2}$ has Pontryagin number $3$ and signature $1$ , related by Hirzebruch's formula $σ = p_{1} /3$ . The number $3$ is the integer-valued bordism fingerprint of $CP^{2}$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ be a closed smooth $n$ -manifold (compact, without boundary). Write $T M$ for its tangent bundle and $[M]_{2} \in H_{n} (M; Z /2)$ for the canonical mod-2 fundamental class. If $M$ is oriented, write $[M] \in H_{n} (M; Z)$ for the integral fundamental class fixed by the orientation.

Definition (Stiefel-Whitney numbers). Let $I = (i_{1}, i_{2}, \dots, i_{k})$ be a partition of $n$ , meaning $i_{j} \geq 1$ and $i_{1} + i_{2} + \dots + i_{k} = n$ . The Stiefel-Whitney number of $M$ with index $I$ is

w_{I} [M] := ⟨ w_{i_{1}} (T M) \cup w_{i_{2}} (T M) \cup \dots \cup w_{i_{k}} (T M), [M]_{2} ⟩ \in Z /2,

where $⟨ -, [M]_{2} ⟩ : H^{n} (M; Z /2) \to Z /2$ is the cap-product evaluation against the mod-2 fundamental class. The collection ${w_{I} [M]}_{I ⊢ n}$ over all partitions $I$ of $n$ is the Stiefel-Whitney number system of $M$ .

Definition (Pontryagin numbers). Assume $M$ is oriented with $dim M = 4 k$ for some $k \geq 1$ . Let $I = (i_{1}, \dots, i_{r})$ be a partition of $k$ . The Pontryagin number of $M$ with index $I$ is

p_{I} [M] := ⟨ p_{i_{1}} (T M) \cup p_{i_{2}} (T M) \cup \dots \cup p_{i_{r}} (T M), [M]⟩ \in Z,

a cap-product of integral cohomology classes summing to degree $4 k = dim M$ . The collection ${p_{I} [M]}_{I ⊢ k}$ over all partitions $I$ of $k$ is the Pontryagin number system of $M$ .

The number of Stiefel-Whitney numbers of $M^{n}$ equals the partition number $p (n)$ ; for Pontryagin numbers of $M^{4 k}$ it equals $p (k)$ .

Counterexamples to common slips [Intermediate+]

Slipping into integer arithmetic for SW numbers. Stiefel-Whitney numbers are valued in $Z /2$ , not $Z$ . The expression $w_{1}^{2} [M] = w_{1} [M]^{2}$ is identically equal to $w_{1} [M]$ mod 2 by Fermat's little theorem, but writing $w_{1}^{2} = 4$ as if it were an integer is a category error.
Forgetting orientation for Pontryagin numbers. The integral fundamental class $[M]$ requires $M$ to be oriented. For an unoriented manifold there is only a mod-2 fundamental class and Pontryagin classes still exist integrally, but no integer-valued pairing is canonically defined.
Mis-counting partitions. The partition number $p (4) = 5$ gives five Stiefel-Whitney numbers on a 4-manifold: $w_{4}$ , $w_{1} w_{3}$ , $w_{2}^{2}$ , $w_{1}^{2} w_{2}$ , $w_{1}^{4}$ . The corresponding Pontryagin-side count on a $16$ -manifold uses $p (4) = 5$ partitions: $p_{4}$ , $p_{1} p_{3}$ , $p_{2}^{2}$ , $p_{1}^{2} p_{2}$ , $p_{1}^{4}$ .

Key theorem with proof [Intermediate+]

Theorem (Thom 1954 — Stiefel-Whitney numbers are unoriented-bordism invariants). Let $M^{n}$ and $N^{n}$ be closed smooth $n$ -manifolds. If there exists a compact smooth $(n + 1)$ -manifold $W$ with $\partial W = M ⊔ N$ , then $w_{I} [M] = w_{I} [N]$ for every partition $I$ of $n$ .

Proof. Suppose $W$ is a bordism between $M$ and $N$ , with $\partial W = M ⊔ N$ . Write $i_{M} : M ↪ W$ and $i_{N} : N ↪ W$ for the boundary inclusions, and let $T W ∣_{M} = i_{M}^{*} T W$ denote the restriction of the tangent bundle of $W$ to $M$ .

The crucial identity is that along the boundary of an $(n + 1)$ -manifold $W$ , the tangent bundle of $W$ restricted to the boundary $\partial W$ decomposes as

T W ∣_{\partial W} = T (\partial W) \oplus \underline{R},

where $\underline{R}$ is the product line bundle along the inward normal direction. By the Whitney sum formula,

w (T W ∣_{\partial W}) = w (T (\partial W)) \cdot w (\underline{R}) = w (T (\partial W)) \cdot 1 = w (T (\partial W)) .

So $w_{i} (T M) = i_{M}^{*} w_{i} (T W)$ for every $i$ , and similarly for $N$ .

Now pair against fundamental classes. The fundamental class of $W$ satisfies $\partial [W, \partial W] = [M]_{2} + [N]_{2}$ in $H_{n} (\partial W; Z /2)$ , where $\partial : H_{n + 1} (W, \partial W; Z /2) \to H_{n} (\partial W; Z /2)$ is the connecting map of the long exact sequence of the pair. Cap-product with a cohomology class $α \in H^{n} (W; Z /2)$ gives the pairing identity

⟨ α ∣_{\partial W}, [M]_{2} + [N]_{2} ⟩ = ⟨ δ α, [W, \partial W]⟩,

where $δ : H^{n} (W; Z /2) \to H^{n + 1} (W, \partial W; Z /2)$ is the connecting map. Take $α = w_{i_{1}} (T W) \cup \dots \cup w_{i_{k}} (T W)$ for a partition $I$ of $n$ .

The connecting map $δ$ raises degree by $1$ . But $α$ lives in $H^{n} (W; Z /2)$ , and the cohomology classes $w_{i} (T W)$ are defined globally on $W$ , so $α$ comes from the global cohomology — meaning its image under $δ$ is the obstruction to extending to a relative cohomology class. The image $δ α \in H^{n + 1} (W, \partial W; Z /2)$ may or may not be zero.

The key step: because $α = i_{W}^{*} \tilde{α}$ for a global class $\tilde{α} = w_{i_{1}} (T W) \cup \dots \cup w_{i_{k}} (T W)$ already defined on $W$ , the connecting map $δ$ applied to the restriction of a global class gives zero by exactness of the long sequence

H^{n} (W, \partial W; Z /2) \to H^{n} (W; Z /2) δ H^{n + 1} (W, \partial W; Z /2) .

Therefore $⟨ α ∣_{\partial W}, [M]_{2} + [N]_{2} ⟩ = 0$ in $Z /2$ , which says $w_{I} [M] + w_{I} [N] = 0$ , equivalently $w_{I} [M] = w_{I} [N]$ . $□$

Bridge. This builds toward 03.06.12 (Unoriented bordism and Thom's theorem), where the converse direction completes the picture: SW numbers are not just invariant but complete, separating distinct bordism classes. The same architecture appears again in 03.06.13 (Oriented bordism and Pontryagin-Thom) for the integral side, where Pontryagin numbers play the role of SW numbers and the orientation requirement enters. The foundational reason is exactly that characteristic classes pair with fundamental classes, and the boundary identity $T W ∣_{\partial W} = T (\partial W) \oplus \underline{R}$ makes this pairing compatible with cobordism. Putting these together, this is exactly the bridge between characteristic-class theory and Thom's spectrum-level identification $Ω_{*}^{O} ≅ π_{*} (M O)$ .

Exercises [Intermediate+]

Exercise 8 (hard, short answer).

Show that for a closed oriented 4-manifold $M$ , the signature $σ (M)$ equals $p_{1} [M] /3$ .

Hint

This is Hirzebruch's signature theorem in dimension 4. Use the $L$ -genus $L_{1} = p_{1} /3$ .

Answer

Hirzebruch's signature theorem (Hirzebruch 1956) states $σ (M) = ⟨ L (T M), [M]⟩$ where $L (T M)$ is the Hirzebruch $L$ -class. The first $L$ -polynomial is $L_{1} = p_{1} /3$ . For a closed oriented 4-manifold, $L (T M)$ truncates to $1 + p_{1} /3$ (no higher Pontryagin classes available in degree $\leq 4$ ). So $σ (M) = ⟨ 1 + p_{1} /3, [M]⟩ = ⟨ p_{1} /3, [M]⟩ = p_{1} [M] /3$ . For $CP^{2}$ with $p_{1} [CP^{2}] = 3$ , this gives $σ (CP^{2}) = 1$ , matching the direct computation from the intersection form on $H^{2} (CP^{2}; Z) = Z$ with form $(x, y) \mapsto x y$ . For the K3 surface with $p_{1} [K 3] = - 48$ , this gives $σ (K 3) = - 16$ , matching the $E_{8} (- 1) \oplus E_{8} (- 1) \oplus 3 H$ form on $H^{2} (K 3; Z) = Z^{22}$ .

Lean formalization [Intermediate+]

lean_status: partial — Mathlib lacks bundled characteristic classes and the integral / mod-2 fundamental class of a closed manifold, so the substantive theorems (well-definedness of SW numbers, integrality of Pontryagin numbers, Thom's bordism invariance) are stated with sorry proof bodies in Codex.Modern.CharClasses.SWPontryaginNumbers.

-- See lean/Codex/Modern/CharClasses/SWPontryaginNumbers.lean.

structure ClosedManifold where
  dim : ℕ
  carrier : Type
  topology : TopologicalSpace carrier

def swNumber (M : ClosedManifold) {n : ℕ} (I : Partition n)
    (h : n = M.dim) : ZMod 2 := 0

theorem swNumber_bordism_invariant
    (M M' : ClosedManifold) (h : M.dim = M'.dim)
    (W : UnorientedBordism M M' h)
    {n : ℕ} (I : Partition n) (hn : n = M.dim) :
    swNumber M I hn = swNumber M' I (hn.trans h) := by
  sorry

The placeholder definitions use Unit-valued cohomology to record the type signature; promoting to lean_status: full requires the upstream Mathlib work catalogued in lean_mathlib_gap. The four Thom theorems (SW invariance, Pontryagin invariance, completeness for unoriented bordism, completeness rationally for oriented bordism) are present as statements with sorry bodies, plus numerical records for $CP^{2}$ , $S^{2} \times S^{2}$ , $RP^{2}$ , K3 used in the worked examples below.

Advanced results [Master]

The seven advanced results below organise around three structural questions: how do Stiefel-Whitney numbers detect unoriented bordism (Thom's mod-2 theorem); how do Pontryagin numbers detect oriented bordism rationally (Thom-Hirzebruch); and how do specific worked examples ( $CP^{2}$ , $S^{2} \times S^{2}$ , K3) realise the formalism.

Stiefel-Whitney numbers $w_{I} [M]$ and unoriented bordism invariance

The mod-2 fundamental class $[M]_{2} \in H_{n} (M; Z /2)$ is canonical: every closed topological $n$ -manifold has one, with no orientation hypothesis required. This is the structural reason Stiefel-Whitney numbers exist in such generality. The bordism-invariance proof of the Key theorem above is the foundational argument: characteristic classes restrict from a bordism to its boundary, and the long-exact-sequence vanishing of the connecting map kills the difference $w_{I} [M] - w_{I} [N]$ .

Theorem (Thom 1954, mod-2 case). The pairing $(M, I) \mapsto w_{I} [M]$ descends to a function on unoriented-bordism classes $Ω_{n}^{O} \to ⨁_{I ⊢ n} Z /2$ . Two closed manifolds $M$ and $N$ are unoriented-bordant iff all their Stiefel-Whitney numbers agree.

The completeness direction (SW numbers separate bordism classes) is the deeper half. Thom's proof identifies the unoriented-bordism ring with the homotopy groups of the universal Thom spectrum: $Ω_{*}^{O} ≅ π_{*} (M O)$ . Computing $π_{*} (M O)$ as a polynomial ring $Z /2 [x_{2}, x_{4}, x_{5}, x_{6}, x_{8}, \dots]$ with one generator in each degree $i \neq = 2^{k} - 1$ , and verifying that Stiefel-Whitney numbers form a complete set of mod-2 homotopy invariants of $M O$ , gives the classification: the SW-number map $Ω_{n}^{O} \to ⨁_{I ⊢ n} Z /2$ is injective on bordism classes.

A useful corollary: if all SW numbers of $M^{n}$ vanish, then $M$ is null-bordant. Examples: $Ω_{1}^{O} = Ω_{3}^{O} = 0$ , every closed 1- or 3-manifold is null-bordant. The first non-zero $Ω_{n}^{O}$ for $n \geq 1$ is $Ω_{2}^{O} = Z /2$ , generated by $RP^{2}$ .

Pontryagin numbers $p_{I} [M]$ for orientable 4k-manifolds

The integral side requires orientation. For a closed oriented $4 k$ -manifold $M$ , Pontryagin numbers form a function $Ω_{4 k}^{S O} \to ⨁_{I ⊢ k} Z$ . The bordism-invariance proof mirrors the SW proof, with $Z /2$ replaced by $Z$ and the mod-2 fundamental class replaced by the integral fundamental class fixed by the orientation.

The signs in $p_{I} [M]$ matter: reversing the orientation of $M$ reverses the integral fundamental class, so $p_{I} [\overline{M}] = - p_{I} [M]$ . The Pontryagin classes themselves are independent of orientation (they are defined via complexification, which kills the orientation data — Exercise 7 of 03.06.04); the orientation dependence enters at the fundamental-class evaluation step.

Theorem (integrality). For any oriented closed $4 k$ -manifold $M$ and any partition $I$ of $k$ , the Pontryagin number $p_{I} [M]$ is an integer.

This is immediate from the definitions: $p_{i} (T M) \in H^{4 i} (M; Z)$ are integral classes, the cup-product preserves integrality, and the integral fundamental-class pairing $H^{4 k} (M; Z) \to Z$ produces an integer. The more refined integrality statements — that specific combinations of Pontryagin numbers are divisible by larger integers — are downstream of this basic integrality and feed into the Â-genus integrality (Atiyah-Singer) and the signature divisibility for spin manifolds (Rokhlin's theorem).

The bordism-invariance theorem — Thom 1954

The unoriented case was stated above; the oriented case adds an orientation hypothesis throughout. The proof architecture is identical: the boundary identity $T W ∣_{\partial W} = T (\partial W) \oplus \underline{R}$ extends to oriented manifolds, and the integral long exact sequence

H^{n} (W, \partial W; Z) \to H^{n} (W; Z) δ H^{n + 1} (W, \partial W; Z)

provides the same vanishing of the connecting map applied to a Pontryagin monomial. The conclusion is that $p_{I} [M] = p_{I} [N]$ whenever $M$ and $N$ are oriented-bordant.

Theorem (Thom-Hirzebruch). After tensoring with $Q$ , the rational Pontryagin numbers form a complete set of invariants of oriented bordism in dimension $4 k$ . The signature $σ : Ω_{4 k}^{S O} \to Z$ factors through a specific rational linear combination $L_{k}$ of Pontryagin numbers (the $L$ -genus): $σ (M) = ⟨ L (T M), [M]⟩$ .

The signature theorem (Hirzebruch 1956) gives the explicit formulae:

L_{1} = \frac{1}{3} p_{1}, L_{2} = \frac{1}{45} (7 p_{2} - p_{1}^{2}), L_{3} = \frac{1}{945} (62 p_{3} - 13 p_{2} p_{1} + 2 p_{1}^{3}),

with higher $L_{k}$ given by the multiplicative-sequence machinery of 03.06.15. The integrality of the signature constrains the rational Pontryagin numbers — for instance, $7 p_{2} - p_{1}^{2}$ must be divisible by $45$ on a closed oriented 8-manifold.

Worked examples — $CP^{2}$ , $S^{2} \times S^{2}$ , Kummer surface, K3

$CP^{2}$ . With $H^{*} (CP^{2}; Z) = Z [x] / (x^{3})$ and the Euler sequence giving $c (T CP^{2}) = (1 + x)^{3}$ , the Pontryagin class is $p_{1} (T CP^{2}) = c_{1}^{2} - 2 c_{2} = 9 x^{2} - 6 x^{2} = 3 x^{2}$ (using $c_{1} = 3 x$ , $c_{2} = 3 x^{2}$ ). The Pontryagin number is $p_{1} [CP^{2}] = 3$ and the signature is $σ (CP^{2}) = 1$ , consistent with $σ = p_{1} /3$ .

$S^{2} \times S^{2}$ . The tangent bundle is the external direct sum, so all Pontryagin classes vanish (sphere tangent bundles are stably product). Hence $p_{1} [S^{2} \times S^{2}] = 0$ and $σ (S^{2} \times S^{2}) = 0$ . The 4-manifold is rationally null-bordant.

Kummer surface. The Kummer surface $K = T^{4} / ⟨ ι ⟩$ , the quotient of a 4-torus by the involution $ι : x \mapsto - x$ (resolved at the 16 fixed points), is a smooth 4-manifold diffeomorphic to a K3 surface. Its Euler characteristic is $χ (K) = 24$ and its signature is $σ (K) = - 16$ .

K3 surface. A K3 surface is a closed simply-connected complex surface with vanishing canonical bundle. Its tangent bundle has Pontryagin number $p_{1} [K 3] = - 48$ , and Hirzebruch's signature theorem gives $σ (K 3) = p_{1} [K 3] /3 = - 16$ . The intersection form on $H^{2} (K 3; Z) = Z^{22}$ is $E_{8} (- 1) \oplus E_{8} (- 1) \oplus 3 H$ where $H$ is the hyperbolic lattice $(0110)$ , with signature $(- 8) + (- 8) + 0 = - 16$ . The Euler characteristic $χ (K 3) = 24$ and the Euler-Poincaré formula combine to give the second Chern number $c_{2} [K 3] = χ (K 3) = 24$ .

Higher-dimensional examples — Milnor exotic 7-sphere

In dimension 8, Milnor's discovery of exotic 7-spheres rests on Pontryagin numbers. Construct a closed oriented 8-manifold $W^{8}$ that bounds an exotic 7-sphere $Σ^{7}$ . If $Σ$ were the standard 7-sphere, then $W$ would be null-bordant after capping off, forcing certain Pontryagin-number constraints to hold integrally. Milnor 1956 showed that the Pontryagin number $p_{1}^{2} [W]$ violates the required divisibility, so $Σ$ cannot be the standard sphere — it is an exotic smooth structure on $S^{7}$ . This argument is one of the foundational uses of Pontryagin numbers in differential topology, opening the entire study of exotic differentiable structures and the subsequent classification of homotopy spheres by Kervaire-Milnor.

Universal characteristic-number map and the cobordism ring

The Stiefel-Whitney-number map and the Pontryagin-number map together with the Euler-characteristic map fit into a single homomorphism of graded rings

ϕ : Ω_{*}^{S O} \otimes Q \to Q [p_{1}, p_{2}, \dots],

assigning to each oriented-bordism class its rational Pontryagin-number vector. Thom-Hirzebruch states this is an isomorphism. The unoriented case reads

ϕ_{2} : Ω_{*}^{O} \to Z /2 [w_{2}, w_{4}, w_{5}, w_{6}, w_{8}, \dots],

assigning to each unoriented-bordism class its Stiefel-Whitney-number vector — and Thom shows this is an isomorphism integrally. So Stiefel-Whitney and Pontryagin numbers are exactly the right invariants to detect bordism: they parametrise the cobordism ring.

Synthesis. Stiefel-Whitney and Pontryagin numbers are the foundational reason that characteristic classes interact with bordism: every top-degree characteristic monomial pairs with the fundamental class to produce a number, and the boundary identity $T W ∣_{\partial W} = T (\partial W) \oplus \underline{R}$ propagates this pairing across cobordisms. The central insight is that these numbers are not just bordism invariants but a complete set of bordism invariants (mod 2 for unoriented, rationally for oriented). The bridge is exactly Thom's identification $Ω_{*}^{O} ≅ π_{*} (M O)$ and $Ω_{*}^{S O} \otimes Q ≅ Q [p_{1}, p_{2}, \dots]$ , which identifies the cobordism ring with a polynomial ring whose coordinates are the characteristic numbers.

The structural pattern generalises across the entire characteristic-class apparatus. Putting these together with Hirzebruch's signature theorem, the rational Pontryagin numbers carry not only the bordism information but also the signature, the Â-genus, the Todd genus, and the elliptic genus — every multiplicative genus is a specific rational polynomial in Pontryagin numbers. This is exactly the splitting of the cobordism ring into characteristic-class coordinates that allows Atiyah-Singer to express analytic indices as topological integrals; the pattern recurs in every index-theoretic computation in the curriculum. Appears again in 03.06.15 and 03.09.10 where the same characteristic numbers measure the index of elliptic operators.

Full proof set [Master]

Proposition 1 (boundary identity). Let $W$ be a smooth compact $(n + 1)$ -manifold with boundary $\partial W$ . Then $T W ∣_{\partial W} ≅ T (\partial W) \oplus \underline{R}$ as smooth real vector bundles over $\partial W$ .

Proof. Pick a smooth collar $c : \partial W \times [0, 1) ↪ W$ identifying a neighbourhood of $\partial W$ with $\partial W \times [0, 1)$ . The inward-pointing unit normal vector field along $\partial W$ defines a non-vanishing section of $T W ∣_{\partial W}$ in the direction transverse to $\partial W$ . This section spans a rank-1 product subbundle $\underline{R} \subset T W ∣_{\partial W}$ , and the orthogonal complement (using any Riemannian metric on $W$ ) is identified with $T (\partial W)$ via the collar. The collar identification respects the smooth structure, so $T W ∣_{\partial W} = T (\partial W) \oplus \underline{R}$ as a smooth real vector bundle. $□$

Proposition 2 (bordism invariance of SW numbers). Let $W^{n + 1}$ be a smooth compact $(n + 1)$ -manifold with $\partial W = M ⊔ N$ . Then $w_{I} [M] = w_{I} [N]$ for every partition $I$ of $n$ .

Proof. Apply Proposition 1 along the boundary $\partial W = M ⊔ N$ . The Whitney sum formula and the vanishing of Stiefel-Whitney classes for product bundles gives $w (T (\partial W)) = w (T W ∣_{\partial W})$ , so $w_{i} (T M) = i_{M}^{*} w_{i} (T W)$ and $w_{i} (T N) = i_{N}^{*} w_{i} (T W)$ for all $i$ .

For a partition $I$ of $n$ , let $α = w_{i_{1}} (T W) \cup \dots \cup w_{i_{k}} (T W) \in H^{n} (W; Z /2)$ . The fundamental-class identity for manifolds with boundary states $\partial [W, \partial W]_{2} = [\partial W]_{2} = [M]_{2} + [N]_{2}$ in $H_{n} (\partial W; Z /2)$ , where $\partial : H_{n + 1} (W, \partial W; Z /2) \to H_{n} (\partial W; Z /2)$ is the connecting homomorphism.

The cap-product compatibility gives

⟨ i_{\partial W}^{*} α, [M]_{2} + [N]_{2} ⟩ = ⟨ α, i_{\partial W *} [\partial W]_{2} ⟩ = ⟨ α, \partial [W, \partial W]_{2} ⟩ = ⟨ δ α, [W, \partial W]_{2} ⟩,

with $δ : H^{n} (W; Z /2) \to H^{n + 1} (W, \partial W; Z /2)$ the long-exact-sequence connecting map. The crucial fact is that the long exact sequence of the pair

\dots \to H^{n} (W, \partial W; Z /2) j^{*} H^{n} (W; Z /2) δ H^{n + 1} (W, \partial W; Z /2) \to \dots

is exact at the middle term. Since $α = w_{i_{1}} (T W) \cup \dots$ is the cup-product of classes defined globally on $W$ , it lies in the image of $j^{*}$ if and only if the original tangent bundle $T W$ extends as a relative bundle pair — and this is always the case because $T W$ is defined on all of $W$ . So $α \in im (j^{*}) = ker (δ)$ , giving $δ α = 0$ . Therefore $⟨ i_{\partial W}^{*} α, [M]_{2} + [N]_{2} ⟩ = 0$ , equivalently $w_{I} [M] + w_{I} [N] = 0$ in $Z /2$ , equivalently $w_{I} [M] = w_{I} [N]$ . $□$

Proposition 3 (integrality of Pontryagin numbers). For any oriented closed $4 k$ -manifold $M$ and any partition $I$ of $k$ , the Pontryagin number $p_{I} [M]$ is an integer.

Proof. The Pontryagin classes $p_{i} (T M) \in H^{4 i} (M; Z)$ are integral by construction (as integral lifts of the even Chern classes of the complexified tangent bundle, with the standard sign $p_{i} = (- 1)^{i} c_{2 i} (T M \otimes C)$ ). The cup-product of integral classes is integral, so

p_{i_{1}} (T M) \cup \dots \cup p_{i_{r}} (T M) \in H^{4 k} (M; Z) .

The integral fundamental-class pairing $⟨ -, [M]⟩ : H^{4 k} (M; Z) \to Z$ is a $Z$ -linear map by definition of $[M]$ . The composition produces an integer. $□$

Proposition 4 (bordism invariance of Pontryagin numbers). Let $W^{4 k + 1}$ be a smooth compact oriented $(4 k + 1)$ -manifold with $\partial W = M ⊔ \overline{N}$ , where $\overline{N}$ denotes $N$ with reversed orientation. Then $p_{I} [M] = p_{I} [N]$ for every partition $I$ of $k$ .

Proof. Repeat the proof of Proposition 2 with integral coefficients and Pontryagin classes in place of Stiefel-Whitney classes. The boundary identity (Proposition 1) extends to oriented manifolds; the Whitney sum formula for Pontryagin classes gives $p_{i} (T (\partial W)) = i_{\partial W}^{*} p_{i} (T W)$ ; the long exact sequence of the pair $(W, \partial W)$ at integral coefficients gives $δ β = 0$ for $β = p_{i_{1}} (T W) \cup \dots \cup p_{i_{r}} (T W)$ . The orientation reversal of $N$ ensures $[\partial W] = [M] - [N]$ as oriented fundamental classes (rather than $[M] + [N]$ ), so the conclusion is $p_{I} [M] - p_{I} [N] = 0$ . $□$

Proposition 5 (Hirzebruch signature theorem, dim 4 case). For a closed oriented 4-manifold $M$ , the signature $σ (M)$ equals $p_{1} [M] /3$ .

Proof. The signature is a homomorphism $σ : Ω_{4}^{S O} \to Z$ from the rational oriented-bordism ring; the Pontryagin-number map $p_{1} [\cdot] : Ω_{4}^{S O} \to Z$ is another. Both vanish on the null-bordant class. The rational oriented-bordism ring in dimension 4 is $Ω_{4}^{S O} \otimes Q = Q$ , generated by the class of $CP^{2}$ . So $σ$ and $p_{1} [\cdot] /3$ are both rational linear functionals on a one-dimensional space, agreeing on the generator $[CP^{2}]$ (where both equal $1$ ). They are therefore equal on all of $Ω_{4}^{S O} \otimes Q$ . Since both $σ$ and $p_{1} [\cdot]$ are integral on every closed oriented 4-manifold, and $p_{1} [CP^{2}] /3 = 1 = σ (CP^{2})$ , the identity $σ (M) = p_{1} [M] /3$ holds integrally on all of $Ω_{4}^{S O}$ . $□$

Connections [Master]

Stiefel-Whitney classes 03.06.03. Provides the mod-2 cohomology classes $w_{i} (T M)$ whose top-degree monomials are evaluated against $[M]_{2}$ to produce SW numbers. The unit here is the numerical-invariant downstream specialisation of 03.06.03.
Pontryagin and Chern classes 03.06.04. Provides the integral cohomology classes $p_{i} (T M)$ whose top-degree monomials are evaluated against $[M]$ to produce Pontryagin numbers. The same naturality + Whitney formula structure that defines the classes in 03.06.04 underlies the bordism invariance of their numerical evaluations here.
Chern-Weil homomorphism 03.06.06. Provides curvature representatives for Pontryagin classes via invariant polynomials of the connection on $T M$ . The de Rham incarnation of a Pontryagin number is then the integral $\int_{M}$ of a top-degree polynomial in the curvature.
Unoriented bordism and Thom's theorem 03.06.12. This unit ships the bordism-invariance side; 03.06.12 (pending) ships the completeness side — that SW numbers separate unoriented-bordism classes. Together they identify $Ω_{*}^{O}$ with the polynomial ring in $Z /2$ on generators detected by SW numbers.
Oriented bordism and Pontryagin-Thom 03.06.13. Same architecture as 03.06.12 but for oriented bordism, with Pontryagin numbers in the role of SW numbers and the rationalisation $Ω_{*}^{S O} \otimes Q ≅ Q [p_{1}, p_{2}, \dots]$ as the completeness statement. Wall 1960 sharpened this integrally: the joint family of Stiefel-Whitney and Pontryagin numbers of the present unit classifies $Ω_{*}^{S O}$ up to the residual 2-torsion captured by the Adams spectral sequence. Anchor phrase: Pontryagin + SW numbers classify oriented bordism (Wall 1960).
Hirzebruch signature theorem 03.06.11. The signature $σ (M)$ is a specific rational linear combination of Pontryagin numbers via the $L$ -genus, $σ (M) = ⟨ L (T M), [M]⟩$ . Builds on the integrality + bordism invariance results of this unit.
Multiplicative sequences and L, Â, Todd genera 03.06.15. Every multiplicative genus is a rational polynomial in Pontryagin numbers (real case) or Chern numbers (complex case). Builds toward the index theorem of 03.09.10.
Combinatorial Pontryagin classes and exotic 7-spheres 03.06.17. Milnor's discovery of exotic smooth structures on $S^{7}$ uses Pontryagin numbers and the divisibility constraints from the bordism-invariance theorem proved here. The numerical-invariant machinery is the foundational reason exotic spheres can be detected at all.
Whitney duality and immersion obstructions 03.06.16. Whitney duality $w (T M) \cup w (ν M) = 1$ is exactly the SW-number compatibility statement on the tangent and normal bundles of an immersed manifold: pairing the dual SW classes of 03.06.10 with the fundamental class produces matched numerical invariants whose mismatch obstructs the existence of an immersion in given codimension. Anchor phrase: Whitney duality is the SW-number compatibility result.
Atiyah-Singer index theorem 03.09.10. The analytic index of an elliptic operator on a closed manifold is computed as the integral of a specific characteristic monomial against the fundamental class — exactly a characteristic number. Pontryagin numbers and Chern numbers are the topological side of the index formula.

Historical & philosophical context [Master]

Eduard Stiefel 1936 introduced the Stiefel-Whitney classes in his Zürich thesis ^{[Stiefel1936]}, in the context of obstructions to extending sections of frame bundles on manifolds; Whitney 1937 independently gave the cohomological side and proved the product formula ^{[Whitney1937]}. The numerical pairings $w_{I} [M]$ appeared implicitly in this early work, but the systematic bordism-invariance theory had to wait for René Thom.

René Thom 1954 ^[Thom1954] introduced the bordism ring $Ω_{*}^{O}$ and proved that Stiefel-Whitney numbers separate unoriented-bordism classes, in his foundational paper Quelques propriétés globales des variétés différentiables (Comment. Math. Helv. 28). Thom's argument used the Pontryagin-Thom construction: bordism classes correspond to homotopy classes of maps from a sphere into the Thom spectrum $M O$ , and computing $π_{*} (M O)$ as a polynomial ring identifies SW numbers as a complete set of homotopy invariants.

Friedrich Hirzebruch 1956 ^{[Hirzebruch1956]} developed the parallel oriented theory in Topological Methods in Algebraic Geometry (§1.5 + Ch. 1 Appendix). The signature theorem $σ (M) = ⟨ L (T M), [M]⟩$ identifies the signature as a specific rational combination of Pontryagin numbers. C.T.C. Wall 1960 (Ann. Math. 72) ^[Wall1960] determined the integral oriented-bordism ring $Ω_{*}^{S O}$ , completing the classification programme.

The numerical-invariant philosophy of characteristic numbers was a structural turning point in algebraic topology: cohomology classes are abstract, but their evaluations against fundamental classes are integers (or mod-2 integers), and these integers carry surprisingly complete bordism information. The pattern was abstracted by Atiyah-Singer 1963 into the index-theoretic statement that every analytic index is a characteristic-number integral — connecting Pontryagin numbers directly to spectral data of elliptic operators on $M$ .

Bibliography [Master]

@article{Stiefel1936,
  author = {Stiefel, Eduard},
  title = {Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten},
  journal = {Commentarii Mathematici Helvetici},
  volume = {8},
  year = {1936},
  pages = {305-353},
}

@article{Whitney1937,
  author = {Whitney, Hassler},
  title = {Topological properties of differentiable manifolds},
  journal = {Bulletin of the American Mathematical Society},
  volume = {43},
  year = {1937},
  pages = {785-805},
}

@article{Thom1954,
  author = {Thom, Ren\'e},
  title = {Quelques propri\'et\'es globales des vari\'et\'es diff\'erentiables},
  journal = {Commentarii Mathematici Helvetici},
  volume = {28},
  year = {1954},
  pages = {17-86},
}

@book{Hirzebruch1956,
  author = {Hirzebruch, Friedrich},
  title = {Topological Methods in Algebraic Geometry},
  publisher = {Springer},
  year = {1956},
  edition = {3rd},
}

@article{Wall1960,
  author = {Wall, C. T. C.},
  title = {Determination of the cobordism ring},
  journal = {Annals of Mathematics},
  volume = {72},
  year = {1960},
  pages = {292-311},
}

@article{Milnor1956,
  author = {Milnor, John},
  title = {On manifolds homeomorphic to the 7-sphere},
  journal = {Annals of Mathematics},
  volume = {64},
  year = {1956},
  pages = {399-405},
}

@book{MilnorStasheff1974,
  author = {Milnor, John and Stasheff, James},
  title = {Characteristic Classes},
  publisher = {Princeton University Press},
  year = {1974},
}

@book{Stong1968,
  author = {Stong, Robert},
  title = {Notes on Cobordism Theory},
  publisher = {Princeton University Press},
  year = {1968},
}

Cycle 5 unit, math T1 frontier, characteristic-classes cluster — closes the bridge between Stiefel-Whitney/Pontryagin classes and Thom's bordism classification.

Prerequisites

03.06.03
03.06.04
03.06.06
03.02.01
03.12.11

Tier anchors

beginner: Milnor-Stasheff §4 + §16 informal level; visual count-the-monomial-evaluations picture
intermediate: Milnor-Stasheff §4 (SW numbers), §16 (Pontryagin numbers); Hirzebruch §1.5
master: Milnor-Stasheff §4 + §16 + §17 + Appendix A; Thom 1954 Comment. Math. Helv. 28; Hirzebruch 1956 Topological Methods §1.5

References

TODO_REF
Milnor-Stasheff — Characteristic Classes · §4 Stiefel-Whitney numbers, §16 Pontryagin numbers, §17 bordism applications
TODO_REF
Thom — Quelques propriétés globales des variétés différentiables · Comment. Math. Helv. 28 (1954), 17–86
TODO_REF
Hirzebruch — Topological Methods in Algebraic Geometry · §1.5 Pontryagin numbers and the signature theorem
TODO_REF
Stong — Notes on Cobordism Theory · Ch. III–IV, unoriented and oriented bordism
TODO_REF
Wall — Determination of the cobordism ring · Ann. Math. 72 (1960), 292–311

Lean module

Codex.Modern.CharClasses.SWPontryaginNumbers

Mathlib gap

Mathlib has neither bundled characteristic classes of vector bundles
(Stiefel-Whitney, Pontryagin, Chern, Euler) nor the integral / mod-2
fundamental class of a closed manifold, nor Thom's bordism ring
spectrum MO/MSO. A formalisation needs `w_i(TM) ∈ H^i(M; ZMod 2)`,
`p_i(TM) ∈ H^{4i}(M; ℤ)`, the cap-product evaluation against the
fundamental class, and Thom's theorem identifying unoriented and
oriented bordism with the homotopy groups of MO and MSO.

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 45m
master: 90m