The Maps BSp(1) →BSp(n)

Abstract

Let Sp(n) be the symplectic Lie group. Then it is known that given a map f: BSp(l) -BSp(l), f*: H4(BSp(1), Z)H4(BSp(l), Z) is zero or multiplication by the square of an odd integer. We generalise the latter part of this result using symplectic K*-theory. We begin with some notation. Let T' C Sp(n) be the standard maximal torus [2] and BSp(n) a classifying space [8]. All cohomology will have integer coefficients. From [5], we have H (BT') Z[t. t I, dim ti 2. The inclusion T' C Sp(n) induces an injection of H*(BSp(n)) onto the Weyl group invariants in H*(BT '). Let T C Sp(1) be the standard maximal torus. Then we extract the following from [3]. Proposition 1. If f: BSp(1) BSp(n) is a map and f* H*(BSp(n)) H*(BSp(1)), then there is an extension 0*: H*(BTI) H*(BT) of f *. Thus, if H (BT) Z[t], then O*ti = m(i)t for some integer m(i). In this note we prove the following: In the set lm(1), m(2), .. , m(n)}, each even m(i) occurs an even number of times. For this purpose we compute f !: KU0(BSp(n)) KU0(BSp(1)), where KU* is complex K*-theory. From [4] we find that KU0(BT') Z[[s1, . .. , sn]] where (1 + si) is the virtual canonical line bundle over BS1. Put Z = 1 + si. KU0(BSp(n)) is isomorphic to the Weyl group invariants in KU0(BT') [4], and the Weyl group acts by permuting the zi and inverting: zi z. Hence KU?3(BSp)) M[[y1X ... X 7 Yn]] Yi =ith elementary symmetric function in (zj + z12). For BSp(1), put y1 = y. Let G be a compact connected Lie group and R(G) its complex representation ring. Then in [4, p. 29], an isomorphism a3: R(G) KU0(BG) is described (here R(G) is the completion of R(G) under the augmentation topology). There are also monomorphisms a: R(G) -KU?(BG) and R(G) R(G). If Sp and U are the big symplectic and unitary groups, the standard inclusion 1: Sp U induces a monomorphism 1*: KSp*(BSp(n)) KU*(BSp(n)) of abelian groups. An element of KU*(BSp(n)) is called symplectic if it is in the image of 1*. Received by the editors July 23, 1974. AMS (MOS) subject classifications (1970). Primary 55F40; Secondary 55F50.