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.

Contenuti 1 Enunciato del teorema 2 Sketch of proof of the signature theorem 3 generalizzazioni 4 Riferimenti 5 Sources Statement of the theorem The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series {stile di visualizzazione {x over tanh(X)}=somma _{kgq 0}{{2^{2K}B_{2K} Sopra (2K)!}x^{2K}}=1+{x^{2} Sopra 3}-{x^{4} Sopra 45}+cdot .} The first two of the resulting L-polynomials are: {stile di visualizzazione L_{1}={tfrac {1}{3}}p_{1}} {stile di visualizzazione L_{2}={tfrac {1}{45}}(7p_{2}-p_{1}^{2})} By taking for the {stile di visualizzazione p_{io}} the Pontryagin classes {stile di visualizzazione p_{io}(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, {stile di visualizzazione [M]} , è uguale a {displaystyle sigma (M)} , the signature of M (cioè. the signature of the intersection form on the 2nth cohomology group of M): {displaystyle sigma (M)=langle L_{n}(p_{1}(M),punti ,p_{n}(M)),[M]sonaglio .} 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 {stile di visualizzazione Omega _{*}^{testo{SO}}otimes mathbb {Q} } è uguale a {stile di visualizzazione Omega _{*}^{testo{SO}}otimes mathbb {Q} = matematica bb {Q} [mathbb {P} ^{2}(mathbb {C} ),mathbb {P} ^{4}(mathbb {C} ),ldot ],} the polynomial algebra generated by the oriented cobordism classes {stile di visualizzazione [mathbb {P} ^{2io}(mathbb {C} )]} of the even dimensional complex projective spaces, it is enough to verify that {displaystyle sigma (mathbb {P} ^{2io})=1=langle L_{io}(p_{1}(mathbb {P} ^{2io}),ldot ,p_{n}(mathbb {P} ^{2io})),[mathbb {P} ^{2io}]sonaglio } per tutti 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. I classici in matematica. Translation from the German and appendix one by R. l. e. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. stampa. of the 3rd ed.). Berlino: Springer-Verlag. ISBN 3-540-58663-6. Sources F. Hirzebruch, The Signature Theorem. Reminiscences and recreation. Prospects in Mathematics, Annals of Mathematical Studies, Gruppo musicale 70, 1971, S. 3–31. Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annali di studi matematici. Stampa dell'Università di Princeton; University of Tokyo Press. ISBN 0-691-08122-0. Categorie: Theorems in algebraic topologyTheorems in differential topology