Hirzebruch signature theorem

In differential topology, an area of mathematics, the Hirzebruch signature theorem^[1] (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combination of Pontryagin numbers called the L-genus. It was used in the proof of the Hirzebruch–Riemann–Roch theorem.

Statement of the theorem

The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series

${\displaystyle {x \over \tanh(x)}=\sum _{k\geq 0}{{2^{2k}B_{2k} \over (2k)!}x^{2k}}=1+{x^{2} \over 3}-{x^{4} \over 45}+\cdots .}$

The first two of the resulting L-polynomials are:

• ${\displaystyle L_{1}={\tfrac {1}{3}}p_{1}}$
• ${\displaystyle L_{2}={\tfrac {1}{45}}(7p_{2}-p_{1}^{2})}$

(for further L-polynomials see ^[2] or OEIS: A237111).

By taking for the ${\displaystyle p_{i}}$ the Pontryagin classes ${\displaystyle p_{i}(M)}$ of the tangent bundle of a 4n dimensional smooth closed oriented manifold M one obtains the L-classes of M. Hirzebruch showed that the n-th L-class of M evaluated on the fundamental class of M, ${\displaystyle [M]}$, is equal to ${\displaystyle \sigma (M)}$, the signature of M (i.e. the signature of the intersection form on the 2nth cohomology group of M):

${\displaystyle \sigma (M)=\langle L_{n}(p_{1}(M),\dots ,p_{n}(M)),[M]\rangle .}$

Sketch of proof of the signature theorem

René Thom had earlier proved that the signature was given by some linear combination of Pontryagin numbers, and Hirzebruch found the exact formula for this linear combination by introducing the notion of the genus of a multiplicative sequence.

Since the rational oriented cobordism ring ${\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q} }$ is equal to

${\displaystyle \Omega _{*}^{\text{SO}}\otimes \mathbb {Q} =\mathbb {Q} [\mathbb {P} ^{2}(\mathbb {C} ),\mathbb {P} ^{4}(\mathbb {C} ),\ldots ],}$

the polynomial algebra generated by the oriented cobordism classes ${\displaystyle [\mathbb {P} ^{2i}(\mathbb {C} )]}$ of the even dimensional complex projective spaces, it is enough to verify that

${\displaystyle \sigma (\mathbb {P} ^{2i})=1=\langle L_{i}(p_{1}(\mathbb {P} ^{2i}),\ldots ,p_{n}(\mathbb {P} ^{2i})),[\mathbb {P} ^{2i}]\rangle }$

for all i.

Generalizations

The signature theorem is a special case of the Atiyah–Singer index theorem for the signature operator. The analytic index of the signature operator equals the signature of the manifold, and its topological index is the L-genus of the manifold. By the Atiyah–Singer index theorem these are equal.