On unstable complex James numbers

One of the fundamental problems in algebraic topology is the determinination of the homotopy groups of topological spaces. The homotopy groups of the classical Lie groups and their coset spaces play a central role in the theory. The calculation of the James numbers is intimately related to the determination of the homotopy groups of the unitary groups, which in turn are used to prove certain existence theorems in topology and differential geometry.; Let {dollar}Wsb{lcub}n,k{rcub}{dollar} be the complex Stiefel manifold of all orthonormal k-frames in {dollar}Csp{lcub}n{rcub}{dollar}. Denote by p: {dollar}Wsb{lcub}n,k{rcub}to Ssp{lcub}2n-1{rcub}{dollar} the fibre bundle projection mapping each k-frame to its last vector. The unstable complex James number {dollar}bsb{lcub}n,k{rcub}{dollar} is defined as the index of {dollar}p*sppi sb{lcub}2n-1{rcub}(Wsb{lcub}n,k{rcub}){dollar} in {dollar}pisb{lcub}2n-1{rcub}(Ssp{lcub}2n-1{rcub}).{dollar}; This paper gives completed calculations of the numbers {dollar}bsb{lcub}n,n-4{rcub}{dollar}, and partial results on {dollar}bsb{lcub}n,k{rcub}{dollar}, for {dollar}kle n-5.{dollar} In particular, {dollar}bsb{lcub}n,n-5{rcub}{dollar} has been determined up to a factor dividing 4.; Our calculations are patterned on Walker's calculations of {dollar}bsb{lcub}n,n-3{rcub}{dollar}. We calculate lower bounds using number theoretic methods, and upper bounds using topological methods. Our major tools include Toda brackets, K-theory, and the Adams e-invariant.; Our keystone technique for establishing upper bounds involves Oka's splitting{dollar}{dollar}pisb{lcub}i{rcub}(SU(p + m);p)cong sumsbsp{lcub}j=1{rcub}{lcub}m{rcub} pisb{lcub}i{rcub}(Bsb{lcub}j{rcub}(p);p) + sumsbsp{lcub}j=m+1{rcub}{lcub}p-1{rcub} pisb{lcub}i{rcub}(Ssp{lcub}2j+1{rcub};p){dollar}{dollar}and the fibration homotopy sequence {dollar}{dollar}longrightarrowpisb{lcub}i{rcub}(Ssp{lcub}2j+1{rcub};p){lcub}buildrel{lcub}isb*{rcub}over{lcub}longrightarrow{rcub}{rcub}pisb{lcub}i{rcub}((Bsb{lcub}j{rcub}(p);p) {lcub}buildrel{lcub}jsb*{rcub}over{lcub}longrightarrow{rcub}{rcub}pisb{lcub}i{rcub}(Ssp{lcub}2p+2j-1{rcub};p) {lcub}buildrel{lcub}partial{rcub}over{lcub}longrightarrow{rcub}{rcub}.{dollar}{dollar}We show the existence of an element {dollar}gammainpisb{lcub}2n-2{rcub}(Ssp{lcub}2k-1{rcub}){dollar} of appropriate order which is mapped by iterated suspensions to a generator of {dollar}Im(J)subsetpisbsp{lcub}2n-2 {rcub}{dollar} and for which {dollar}partial(gamma)=0{dollar}. We pick {dollar}lbrackgammarbrackinpisb{lcub}2n-2{rcub}(Bsb{lcub}k-pj{rcub}p){dollar} such that {dollar}jsb*(lbrack gammarbrack )=gamma{dollar}. The e-invariant establishes that {dollar}lbrackgammarbrack,{dollar} viewed as an element of{dollar}pisb{lcub}2n-2{rcub}(SU(k)){dollar}, is mapped monomorphically into {dollar}pisb{lcub}2n-2{rcub}(SU(n-1)){dollar} by the map induced by inclusion.