Genus of a multiplicative sequence

In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties. A genus assigns a number to each manifold X such that (where is the disjoint union); if X is the boundary of a manifold with boundary. The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value is in some ring, often the ring of rational numbers, though it can be other rings such as or the ring of modular forms. The conditions on can be rephrased as saying that is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring. Example: If is the signature of the oriented manifold X, then is a genus from oriented manifolds to the ring of integers. Multiplicative sequence A sequence of polynomials in variables is called multiplicative if implies that If is a formal power series in z with constant term 1, we can define a multiplicative sequence by where is the kth elementary symmetric function of the indeterminates . (The variables will often in practice be Pontryagin classes.) The genus of compact, connected, smooth, oriented manifolds corresponding to Q is given by where the are the Pontryagin classes of X. The power series Q is called the characteristic power series of the genus . A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4k for positive integers k, implies that this gives a bijection between formal power series Q with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.

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Hirzebruch signature theorem

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 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. 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: (for further L-polynomials see or ).

Friedrich Hirzebruch

Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as "the most important mathematician in Germany of the postwar period." Hirzebruch was born in Hamm, Westphalia in 1927. His father of the same name was a maths teacher. Hirzebruch studied at the University of Münster from 1945–1950, with one year at ETH Zürich.

Signature (topology)

In the field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension divisible by four. This invariant of a manifold has been studied in detail, starting with Rokhlin's theorem for 4-manifolds, and Hirzebruch signature theorem. Given a connected and oriented manifold M of dimension 4k, the cup product gives rise to a quadratic form Q on the 'middle' real cohomology group The basic identity for the cup product shows that with p = q = 2k the product is symmetric.

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