Hirzebruch signature theorem

Hirzebruch signature theorem 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. It was used in the proof of the Hirzebruch–Riemann–Roch theorem.

Contents 1 Statement of the theorem 2 Sketch of proof of the signature theorem 3 Generalizations 4 References 5 Sources Statement of the theorem The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series {displaystyle {x over tanh(x)}=sum _{kgeq 0}{{2^{2k}B_{2k} over (2k)!}x^{2k}}=1+{x^{2} over 3}-{x^{4} over 45}+cdots .} The first two of the resulting L-polynomials are: {displaystyle L_{1}={tfrac {1}{3}}p_{1}} {displaystyle L_{2}={tfrac {1}{45}}(7p_{2}-p_{1}^{2})} By taking for the {displaystyle p_{i}} the Pontryagin classes {displaystyle p_{i}(M)} 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, {displaystyle [M]} , is equal to {displaystyle sigma (M)} , the signature of M (i.e. the signature of the intersection form on the 2nth cohomology group of M): {displaystyle sigma (M)=langle L_{n}(p_{1}(M),dots ,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 combination by introducing the notion of the genus of a multiplicative sequence.

Since the rational oriented cobordism ring {displaystyle Omega _{*}^{text{SO}}otimes mathbb {Q} } is equal to {displaystyle Omega _{*}^{text{SO}}otimes mathbb {Q} =mathbb {Q} [mathbb {P} ^{2}(mathbb {C} ),mathbb {P} ^{4}(mathbb {C} ),ldots ],} the polynomial algebra generated by the oriented cobordism classes {displaystyle [mathbb {P} ^{2i}(mathbb {C} )]} of the even dimensional complex projective spaces, it is enough to verify that {displaystyle sigma (mathbb {P} ^{2i})=1=langle L_{i}(p_{1}(mathbb {P} ^{2i}),ldots ,p_{n}(mathbb {P} ^{2i})),[mathbb {P} ^{2i}]rangle } 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 ^ 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. Sources F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Band 70, 1971, S. 3–31. Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies. Princeton University Press; University of Tokyo Press. ISBN 0-691-08122-0. Categories: Theorems in algebraic topologyTheorems in differential topology