Localization in equivariant bordisms (Q1321009)

This paper is the first of two articles devoted to a derivation of a universal local formula in smooth \(G\)-equivariant bordism (\(G\) a finite group). Here `local' means based on information derived from neighborhoods of fixed point sets. Such a universal formula efficiently generates specified local formulae for a variety of equivariant genera [the author, Topology 31, No. 4, 713-733 (1992; Zbl 0777.57014)]. Working modulo the class of \(G\)-bordisms induced by the \(H\)-bordisms, \(H\) running over a family of proper subgroups of \(G\), one finds that the \(G\)- bordisms class of \(M\) is congruent to a universal expression in certain standard \(G\)-actions with bordism-valued characteristic classes of the \(G\)-normal bundle \(\nu (M^ G, M)\) as coefficients. The standard actions consist of a specified list of algebraic actions on Riemann surfaces, together with linear actions on complex projective spaces. The derivation of the universal formula breaks up into two main steps; the first one studying the phenomenon of localization in \(G\)-bordism, and the second developing a `geometric' theory of characteristic classes for vector bundles with a group action. The first step is the subject of the paper under review, and includes inter alia an effective computation (inverting certain primes all dividing the group order \([G]\)) of the various bordism theories in classical `nonequivariant' terms. The second step is the subject of the author's previously cited paper. In Section 1 the author reviews some basic notions from equivariant bordism and constructs a `Mayer-Vietoris' spectral sequence. In Section 2 he discusses the effect of algebraic localization on \(G\)-bordism, and in Section 3 the compatibility between algebraic and geometric localization. Section 4 is devoted to the calculation of certain relative bordism groups in non-equivariant classical terms. The geometric methods in the paper should be compared to the homotopical methods of tom Dieck, who works with equivariant homotopy bordism theories based on equivariant spectra.