The Image of H ∗ (BSO; Z 2 ) in H ∗ (BO; Z 2 )

Abstract

We construct explicit polynomial generators of the image of I1*(BSO;Z2) in I1*(BO;Z2). Following the computation of the A-comodule algebra structure of H *(MSO; Z2) by D. Pengelley [6], there has been renewed interest in studying bordism from the homology (as opposed to cohomology) point of view. Recently various families of generators for H (BSO; Z2) have been constructed by Bahri [1], Baker [2], Kochman [4, 5], and Pengelley [6, 7] using a variety of algebraic and geometric methods. In this note we construct polynomial generators of the image of H* (BSO; Z2) in H*(BO; Z2) in terms of the canonical polynomial generators x,, n > l, of H* (BO; Z2) (see [3, p. 479.]). These polynomial generators of H* (BSO; Z2) have the distinctive property that they come from H*(BSO(3) ; Z2). In that sense, they are simpler than the other known sets of generators. In order to state our result we introduce the function T which is defined on positive integers as follows: Let I be a positive integer and I = 2' + t, 0 < t < 2' . We define T(l) = t and and put x0 = 1 . 1. Definition. We define a sequence of elements Y2 , Y3, Y4, ... , y ... of H* (BO; Z2) as follows: (a) If n is a power of 2, then Yn = (xn92). (b) Let n be different than a power of 2 and n = 2 +m *2 + 0 < k 1 < m . Then Yn Xn + X2A Xm2k + E(2kj XlXaXb 2+ <a