Hirzebruch signature theorem explained

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 signatureof a smooth closed oriented manifold by a linear combination of Pontryagin numbers called theL-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 polynomialsassociated to the characteristic power series

{x\over\tanh(x)}=\sum_k\ge{{2^2kB_2k\over(2k)!}x^2k

} = 1 + - +\cdots .

The first two of the resulting L-polynomials are:

L₁=\tfrac13p₁

L₂=\tfrac1{45}(7p₂-

	2)
p
	1

(for further L-polynomials see ^[1] or).

By taking for the

p_i

the Pontryagin classes

p_i(M)

of the tangent bundle of a 4n dimensional smooth closed orientedmanifold 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)=\langleL_n(p_1(M),...,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 combinationby introducing the notion of the genus of a multiplicative sequence.

	SO
\Omega
	*

⊗ \Q

is equal to

	SO
\Omega
	*

⊗ \Q=\Q[P²(\Complex),P⁴(\Complex),\ldots],

the polynomial algebra generated by the oriented cobordism classes

[P²ⁱ(\Complex)]

of the even dimensional complex projective spaces,it is enough to verify that

\sigma(P²ⁱ)=1=\langleL_i(p

	2i

	1(P

),\ldots,

	2i
p
	n(P

)),[P²ⁱ]\rangle

for all i.

Generalizations

The signature theorem is a special case of the Atiyah–Singer index theorem forthe 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.

Sources

F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Band 70, 1971, S. 3–31.
Book: Characteristic classes . Milnor . John W. . Stasheff . James D. . John Milnor . Jim Stasheff . 1974 . Princeton University Press; University of Tokyo Press . 76 . Annals of Mathematics Studies . 0-691-08122-0.

Notes and References

McTague, Carl (2014) "Computing Hirzebruch L-Polynomials".