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Let p be a fixed odd prime. The first author proved in 1989 that for ε = 0 and l, Vin2k-e(SU(n)) has order p'"\ where e(k, n) = min {vp (j\S(kJ)) :n^j with S(kJ) the Stirling number of the second kind. In this paper, we give a more tractable formula for e(k, n) when n < p — p + l by calculating the unstable Novikov spectral sequence. We also determine the abelian group structure when ε = l ; it was known to be cyclic when ε = 0. 1991 Mathematics Subject Classification: 55T15. 1. Statement of results Let p be a fixed odd prime. The (/j-local) t^-periodic homotopy groups, jf π# (Χ), of a space X were defined in [9]. Roughly speaking, they teil the portion of the/j-local homotopy groups of X detected by T-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 v ι -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 A'each ι^-periodic homotopy group ν^π^Χ) is a direct summand of some actual homotopy group nL(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 586 D. Davis, H. Yang (1.1) SU(n 1) -» SU(n) -» S"'. Let v (m) = vp(m) denote the exponent of p in m. Let (1.2) ep(k9n) = mm{vp(/\S(kJ)):n

Tractable formulas for v 1 periodic homotopy groups of SU (n) when n≤p 2 -p + 1