© 1996, Topology Atlas

The v_1-periodic homotopy groups of a space are roughly a localized version of the portion of the p-primary homotopy groups detected by K-theory. The group v_1^{-1}pi_i(X) is defined to be the direct limit over e and k of [M^{i+1+kqp^e}(p^e),X], where q=2(p-1), M^t(n) denotes a Moore space S^{t-1}CUP_ne^t, and the direct system uses Adams maps M^{t+qp^e}(p^e)-> M^t(p^e) and canonical maps M^n(p^{e+1}-> M^n(p^e). If X is a compact space with exponent, v_1^{-1}pi_i(X) is a direct summand of some pi_j(X).

The v_1-periodic homotopy groups of spheres were calculated by Mahowald (p=2) and Thompson (p odd). In [Proc London Math Soc 43 (1991)], Davis observed that, if p is odd, these groups agreed with the 1- and 2-line groups of the unstable Novikov spectral sequence (UNSS), which had been calculated by Bendersky. This implies that the v_1-periodic homotopy groups of spherically resolved spaces can be determined whenever their UNSS can be calculated. This has resulted in the calculation, primarily by Bendersky and Davis, of the v_1-periodic homotopy groups of all compact simple Lie groups except E_8 when p<=5, E_7 when p<= 3, or E_6, F_4, and SO(n) when p=2. It implies results about actual homotopy groups such as that pi_*(SU(n)) contains an element of order p^n-1 for every prime p.