The J Functor and the Nonstable Homotopy Groups of the Unitary Groups

The recent work of Adams, Atiyah, Todd, and Walker on the complex Stiefel manifolds has certain implications regarding the structure of the nonstable homotopy groupsir¡(Um),i>2m. The purpose of this note is to set up some of the machinery for investigating these implications. The information we have obtained so far concerning w^Um), i>2m, is contained in Corollary I, stated below. It is likely that a more refined attack will yield better information. Theorems I and II are here regarded primarily as stepping stones to Corollary I; however, they may prove to be of some intrinsic interest. The general situation we envisage is the following. Let A be a finite C.W. complex and /: Sm—»X a map. There are two cases. 1. The stable homotopy class of the map/is of finite order. In this case, define co-degree/=0. 2. The stable homotopy class of the map/ is of infinite order, i.e., the induced map Hm(Sm; Q)—>Hm(X; Q) is nontrivial. Then, by general theory, the map is 5 rationally co-reducible. That is, there exists an integer k and a map h: Sk(X)-+Sk+m such that the composite hsk(f): Sk+m-+Sk(X)-^Sk+m is an isomorphism on Hm+k(Sk+m; Q). Given such an S rational co-reduction, the map hsk(f) : Sk+m—>Sk+m has a degree which we call, by abuse of notation, deg.(Ä). Among all S co-reductions, there exists one of least positive degree h' (not necessarily unique) and the degree of any other 5 co-reduction is a multiple of deg.(h'). Thus we can make the following definition: co-degree /=deg.(Ä'), h' an 5 rational co-reduction of minimal positive degree. Among the spaces X which fit into the above picture are the complex Grassmanians Vn+k.k= Un+k/UnThere is a standard map S2n+1=Un+i/U„—>Un+k/UnThe co-degree of this map will be denoted by Tl. We have