By taking for the the Pontryagin classes 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, , is equal to , the signature of M
(i.e. the signature of the intersection form on the 2nth cohomology group of M):
Sketch of proof of the signature theoremEdit
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.
the polynomial algebra generated by the oriented cobordism classes
of the even dimensional complex projective spaces,
it is enough to verify that
for all i.
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.
^Hirzebruch, Friedrich (1995) [First published 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.
^McTague, Carl (2014) "Computing Hirzebruch L-Polynomials".
F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Band 70, 1971, S. 3–31.