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The Hirzebruch signature theorem

Statement

Assume AC. For every closed oriented smooth manifold $M$ of dimension $4 k$ , $k \geq 0$ , $σ (M) = ⟨ L_{k} (T M), [M] ⟩ = L [M],$ the Hirzebruch signature theorem. With the signature extended by zero in dimensions not divisible by four, the signature and the L-genus define the same unital $Q$ -algebra homomorphism $Ω_{*}^{S O} \otimes Q \to Q$ . In particular the signature of a closed oriented $4 k$ -manifold is a rational polynomial in its Pontryagin numbers and depends only on the oriented bordism class.

Facts & Assumptions

Given: AC; a closed oriented smooth $4 k$ -manifold $M$ ; the signature $σ$ and the L-genus $L$ of the pair.

[F1]

The signature vanishes on oriented boundaries and is additive and orientation-reversing, hence descends to a well-defined additive map on rationalized oriented bordism classes in each degree $4 k$ (The signature of an oriented boundary vanishes, so the signature is an oriented cobordism invariant, The signature is additive under disjoint union and negates under orientation reversal, The signature of a closed oriented manifold of dimension divisible by four).

[F2]

The L-genus is a unital $Q$ -algebra homomorphism $Ω_{*}^{S O} \otimes Q \to Q$ , additive, multiplicative and vanishing on boundaries (The L-genus is an oriented rational bordism ring homomorphism, The total L-class and the L-genus of a smooth manifold).

[F3]

For $k = 0$ the basis is the positively oriented point. For every $k \geq 1$ the products ${C P}^{2 k_{1}} \times \dots \times {C P}^{2 k_{r}}$ over partitions $k_{1} + \dots + k_{r} = k$ , $k_{i} \geq 1$ , form a $Q$ -basis of $Ω_{4 k}^{S O} \otimes Q$ ; equivalently $Ω_{*}^{S O} \otimes Q$ is the polynomial algebra on the classes $[{C P}^{2 k}]$ , $k \geq 1$ (Products of complex projective spaces span rational oriented bordism, Unoriented and oriented bordism groups, Cartesian product makes bordism a graded ring).

[F4]

On each basis product $P_{J} = {C P}^{2 k_{1}} \times \dots \times {C P}^{2 k_{r}}$ of [F3], with $r \geq 1$ , $k_{i} \geq 1$ and the product of the complex orientations, $σ$ and $L$ both take the value $1$ : $σ (P_{J}) = 1 = L [P_{J}]$ (The signature and the L-genus agree on products of complex projective spaces, The signature and the L-genus agree on complex projective space).

[F5]

The L-genus of a closed oriented $4 k$ -manifold is $L [M] = ⟨ L_{k} (T M), [M] ⟩$ , a rational linear combination of its Pontryagin numbers (The total L-class and the L-genus of a smooth manifold, The Hirzebruch L-polynomials and the total L-class of a real vector bundle).

Proof

technique · direct; compare the two additive functionals on the spanning family of projective-space products

1.1givenF1F2

Both $σ$ and $L$ are well-defined $Q$ -linear functionals on $Ω_{4 k}^{S O} \otimes Q$ : for $σ$ this is the descent statement [F1], extended $Q$ -linearly; for $L$ it is the homomorphism property [F2].

1.2givenF3F4

For $k \geq 1$ , every partition $J$ of $k$ has a nonempty projective-space product, and [F4] gives $σ (P_{J}) = 1 = L [P_{J}]$ . For $k = 0$ the basis is the positively oriented point, and both values are $1$ by the $k = 0$ case of The signature and the L-genus agree on complex projective space.

2.1step 1.1step 1.2F2F3

Equality of functionals: by [F3] the classes $[P_{J}]$ form a $Q$ -basis of $Ω_{4 k}^{S O} \otimes Q$ , and by steps 1.1 and 1.2 the two $Q$ -linear functionals $σ$ and $L$ agree on every basis element; hence $σ = L$ on $Ω_{4 k}^{S O} \otimes Q$ . Since this holds for every $k$ and both functionals are declared zero in dimensions not divisible by four, they agree on all of $Ω_{*}^{S O} \otimes Q$ ; by [F2] and [F3] the common functional is the unital $Q$ -algebra homomorphism recorded in the statement, since it is multiplicative on the polynomial generators according to [F4] and [F2].

3.1step 2.1F1F5∎

Restating: for the closed oriented $4 k$ -manifold $M$ , its class in $Ω_{4 k}^{S O} \otimes Q$ maps to $σ (M)$ under the signature functional and to $L [M] = ⟨ L_{k} (T M), [M] ⟩$ under the L-genus by [F5], and step 2.1 shows these values are equal; the value depends only on the oriented bordism class and is a rational polynomial in the Pontryagin numbers of $M$ by [F5]. The empty manifold and $k = 0$ give the value $1$ on a positively oriented point and $0$ on the empty manifold, consistent with both sides.

Depends on

Used by

Dependency tree · two levels

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Sources

John Milnor and James Stasheff, Characteristic Classes (re-typeset scan; original pagination) (standard reference, not scraped)
Tom Weston, An Introduction to Cobordism Theory (lecture notes, Stanford) (standard reference, not scraped)
Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
Jacob Lurie, The Hirzebruch Signature Formula (Lecture 25, Harvard Math 287x notes) (standard reference, not scraped)