Hirzebruch signature theorem
In differential topology, an area of mathematics, the Hirzebruch signature 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.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:
By taking for the the Pontryagin classes 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,, is equal to, the signature of M
:Sketch of proof of the signature theorem
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 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.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.
