Some remarks on υ₁ -periodic homotopy groups

v1-periodic homotopy groups ofSp(n)

“…1 We employ a similar method, but we use the unstable Novikov spectral sequence (UNSS) to lend precision to the calculations. The p-primary v 1 -periodic homotopy groups of any space X were defined in [15] 1 π n (X) is a finitelygenerated abelian group, which is true in the cases just mentioned. The importance of v −1 1 π * (X) is that it is often computable and yet gives significant information about actual homotopy groups.…”

On the 2-adic $K$-localizations of $H$-spaces

“…Our localization results in this paper will apply to many (but not all) simply-connected finite H-spaces and to related spaces such as the spheres S 4k−1 for k 1. We show that these results allow computations of the v 1 -periodic homotopy groups (see [13], [15]) of our spaces from their united 2-adic K-cohomologies, and thus allow computations of the v 1 -periodic homotopy groups for a large class of simply-connected compact Lie groups from their complex, real, and quaternionic representation theories. The present results will be extended in a which combines the specified cohomologies with the additive operations among them (see Definition 6.1).…”

“…The p-primary v 1 -periodic homotopy groups v −1 1 π * X of a space X at a prime p were defined by Davis and Mahowald [15] and have been studied extensively (see [13]). In this section, we apply the preceding result (Theorem 8.6) on the K/2 * -localizations of our spaces to approach v 1 -periodic homotopy groups at p = 2 using: Definition 9.1 (The functor Φ 1 ).…”

3-primary 𝑣₁-periodic homotopy groups of 𝐹₄ and 𝐸₆

“…The ¿»-primary v\ -periodic homotopy groups of a space X, denoted VyXn*(X ; p), were defined in [15]. They are a localization of the actual homotopy groups, telling roughly the portion which are detected by AT-theory and its operations.…”

“…Toda brackets in vx -periodic homotopy theory may be considered to be ordinary Toda brackets in the mapping telescope described in [15], whose ordinary homotopy groups equal the vx-periodic homotopy groups of the space in question. More explicitly, [3, 3.7] says that a generator of n2x (S7) A cycle representative for this Massey product is given by C -A® 3h4 -3x, where x is defined on S7 and d(x) -A® d(h4).…”