Some Calculations In the Unstable Adams-Novikov Spectral Sequence

Abstract

The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {E r (X)} 9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £ 2 " term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S 2 " +1 .The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E 2 -term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S 2n+1 .We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the Communicated by N.

v1-periodic homotopy groups ofSp(n)

“…For the UNSS, the existence of these exact sequences is proved similarly to the proof for SU(n) in [3], while for the periodic UNSS it follows since the exact sequences commute with the morphisms A of [4, pp. 50-51] which define the direct systems whose direct limit is v…”

On the $p$-Regularlty of Stiefel Manifolds

“…£$•'•" = 0 z/ w 4= 2r n (p -1), w < 2p(p -1) and 0 < t < p. We also need to know whether n^QS 2 "* 1 , S 2n + 1 ) is zero or not in the low dimensional unstable range. The following result is due to Toda (see also [2]). The following result of James is the starting point of our argument.…”

“…In particular, it is p-regular ifn^ (k 2 If follows from the above theorem that, for each odd prime p, almost all ( = except for finitely many undecided cases) W n+ktk and X n+k^k are p-regular if the necessary condition we mentioned above is satisfied. And as a corollary of Theorem 2.17, we show that the attaching maps of the top cells of W n + 2)2 and n + 2,2 produce two families of infinite number of elements of the 2-components of the unstable homotopy groups of spheres, and that the loop space QW n + 2j2 (resp.…”

“…% n +k,k i § p-regular except for the following undecided cases. For p = 13, (n,fc) = (l,6) For p= 19, (n,fc) = (l,9) For p = 23, (n,k) = (l,ll), (2,11), (3,11).…”

Unstable towers in the odd primary homotopy groups of spheres

“…The unstable Novikov spectral sequence. For a simply connected space X there is a spectral sequence (the unstable Novikov spectral sequence) which converges to the homotopy groups of X localized at p with the E2 term depending on the BP homology of X [5,2,3]. The elements described in §1 appear as unstable elements in filtration 2 of this spectral sequence.…”