Pontrjagin numbers of maps

Abstract

We consider an integral valued i£-genus; that is, a ring homomorphism K : Q--»Z which carries the unit of the oriented Thorn bordism ring into 1. The integrality of certain genera, such as the A -genus or the L-genus [6, p. 13], provides some divisibility conditions which must be satisfied by the Pontrjagin numbers of closed manifolds. For the study of the oriented bordism module Q*(X) of a space the Pontrjagin numbers of a map of a closed oriented manifold into X were defined in [4], We shall explore in this note the corresponding divisibility conditions, caused by the integrality of K, which must be satisfied by the Pontrjagin numbers of such maps. For each space X let Q(X) be the collection of all additive rational valued homomorphisms T: Q*(X)—»() which are compatible with the K-genus in the sense that for [ikf, ƒ] G fin(X), [V ] G Qmf T([M,/][FTM]) = (r([Af», f]))(K([V])). The Q(X) is a linear space over the rationals. Let Kj(pi, • • • , pj) be the multiplicative sequence of homogeneous polynomials with rational coefficients which determine K [6, p. 80]. For each closed oriented manifold M, let k(M) G # * ( M n ; Q) be the cohomology class (1, Ki(pi), Ki(pu P2), • • • , Kj(ph ' • * f pj), • * • ), where pjGH*'(M Q) is a Pontrjagin class of the tangent bundle to M. We define a linear homomorphism ¥:H**(X; Q)->Q(X). For c G H**(X; Q) we set TC([M*, ƒ]) = (k(M)f*(c), <r„), where an is the orientation class and the brackets denote the cap product (xnr\k{M )f*{c) followed by the augmentation €*:H*(X; (?)—»(?. A CW-complex X is of finite type if and only if every skeleton X< is finite. We shall only consider such spaces.