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Computing \(v_ 1\)-periodic homotopy groups of spheres and some compact Lie groups. (English) Zbl 0862.55008

James, I. M. (ed.), Handbook of algebraic topology. Amsterdam: North-Holland. 993-1048 (1995).
This is a very useful overview article on the methods and results of the part of unstable homotopy theory dealing with \(v_1\)-periodic homotopy. The \(v_1\)-periodic homotopy groups of a space \(X\), \(v_1^{-1} \pi_*(X)\), are periodic versions of the part of \(\pi_* (X)\) detectable by \(K\)-theory and its operations. They are the second in a hierarchy of theories \(v_n^{-1} \pi_*(X)\), which starts with rational homotopy, and constitute the unstable counterparts of the chromatic pieces in stable homotopy. With mod \(p\) coefficients \(v_1^{-1} \pi_* (X; \mathbb{Z}/p \mathbb{Z})\) is defined simply by inverting the Adams map \(A\), which is a self-map between suspensions the mod \(p\) Moore space inducing an isomorphism in \(K\)-theory. Since usually each group \(v_1^{-1} \pi_*(X)\) is a direct summand of some \(\pi_{i+n} (X)\) for \(n\) large, they actually give information on instable homotopy.
After a careful definition of \(v_1^{-1} \pi_* (X)\), a new and easier accessible computation of \(v_1^{-1} \pi_*(S^{2n+1})\) is given. The computation of \(v_1^{-1} \pi_* (S^{2n+1})\) at the prime \(p\) is reduced to the calculation of the stable groups \(v_1^{-1} \pi_*^S (B^{qn})\), where \(B^{qn}\) is a skeleton of the \(p\)-localization of the classifying space of the symmetric group \(\Sigma_p\). Then it is shown how to compute in general the stable groups \(v_1^{-1} \pi_*^S (X)\) for a spectrum \(X\). They are given by \(v_1^{-1} J_*(X)\), with \(J\) the \(\text{Im} (J)\)-theory spectrum, thus essentially by \(K\)-theory. The two main methods used for computing \(v_1^{-1} \pi_* (X)\) are \(J\)-homology and the unstable Novikov spectral sequence. Both are discussed in detail and illustrated by various examples, e.g. \(\text{SU} (n)\), \(G_2\), \(F_4/G_2\).
For the entire collection see [Zbl 0824.00017].

MSC:

55Q45 Stable homotopy of spheres
57T20 Homotopy groups of topological groups and homogeneous spaces
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