Forum Mathematicum

Forum Math. 8 (1996), 585-619 Forum

Mathematicum

© de Gruyter 1996

Tractable formulas for D|-periodic homotopy groups of SU(n) when n<p^ -p + l

Donald M. Davis and Huajian Yang

( Communicated by Frederick R. Cohen)

Abstract . Let /? be a fixed odd prime. The first author proved in 1989 that for e = 0 and 1,

vï^n2k - t { ^U { n ) ) has order

pe { k , n ) ^ where e{k, n) = m\n{Vp{j\S{kJ)) :n^j<k},

with S(kj') the Stirling number of the second kind. In this paper, we give a more tractable formula for e(Â:, n) when n <p^ —p + lby calculating the unstable Novikov spectral sequence. We also determine the abeUan group structure when e == 1 ; it was known to be cyclic when e = 0.

1991 Mathematics Subject Classification: 55T15.

1 . Statement of results

Letp be a fixed odd prime. The (pAocal) t;i-periodic homotopy groups, v^^n^ (ЛГ), of a space Z were defined in [9]. Roughly speaking, they tell the portion of the/?-local homotopy groups of X detected by ÄT-theory. They are the second of an envisioned sequence of periodic homotopy theories, rational homotopy being the first, which would give a global description of homotopy theory.

The i;i-periodic homotopy groups are particularly nice for spherically resolved spaces. A space is spherically resolved if it can be built from a finite number of spheres and their iterated loop spaces by fibrations. One property which will be important to us is that for a spherically resolved space ЛГеасЬ üj-periodic homotopy group vî^TtiiX) is a direct summand of some actual homotopy group ni{X). We will focus on the case where X is one of the special unitary groups SU{n), which are spherically resolved by the fibrations