Stable summands ofU(n)

Abstract

SynopsisThe special unitary group SU(n) has the stable homotopy type of a wedge of n − 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whether n is finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 when n is finite, but splits into a wedge of two pieces when n is infinite.

Spaces of commuting elements in SU(2)

“…The space Hom(Z T , U(1)) is, of course, a torus of rank l = #T and the stable decomposition of Hom(Z T , U(1)) + as a wedge of 2 l spheres is elementary. But even in this case the S(T )-equivariant decomposition is of interest and was used in [5] to give a new proof of Miller's stable splitting [6] of the unitary group U(l).…”