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.
The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series The first two of the resulting L-polynomials are: (for further L-polynomials see [2] or OEIS: A237111).
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,
, the signature of M (i.e. the signature of the intersection form on the 2nth cohomology group of M): 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.
is equal to 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 analytic index of the signature operator equals the signature of the manifold, and its topological index is the L-genus of the manifold.