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 compact 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

{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:

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

By taking for the $p_{i}$ the Pontryagin classes $p_{i}(M)$ of the tangent bundle of a 4n dimensional smooth compact and 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, $[M]$ , is equal to $\sigma (M)$ , the signature of M (i.e. the signature of the intersection form on the 2nth cohomology group of M ):

\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 $\Omega _{*}^{\text{SO}}\otimes \mathbb {Q}$ is equal to

\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 $[\mathbb {P} ^{2i}(\mathbb {C} )]$ of the even dimensional complex projective spaces, it is enough to verify that

\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.

References

Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-58663-6.

F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Band 70, 1971, S. 3–31.
Milnor John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies. Princeton, New Jersey; Tokyo: Princeton University Press / University of Tokyo Press. ISBN 0-691-08122-0.

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[H1-1] Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-58663-6.