“Hirzebruch signature theorem”的意思、由来-开放百科全书

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Hirzebruch signature theorem

释义

Statement of the theorem
Sketch of proof of the signature theorem
Generalizations
References

In differential topology, an area of mathematics, the Hirzebruch signature theorem (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

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

(i.e. the signature of the intersection form on the 2nth cohomology group of M ):

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 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.

References

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F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Band 70, 1971, S. 3–31.
{{cite book

|author= Milnor John W.
|author2=Stasheff, James D. |authorlink2=Jim Stasheff
|title= Characteristic classes
|work= Annals of Mathematics Studies
|issue=76
|publisher=Princeton University Press / University of Tokyo Press
|location=Princeton, New Jersey; Tokyo
|year= 1974
|isbn= 0-691-08122-0}}

2 : Theorems in differential topology|Theorems in algebraic topology

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