Adams \(e\)-invariant, Toda bracket and \([X,U(n)]\)
Let \(U(n)\) be the unitary group and \(X\) be a finite CW-complex. The homotopy set \(U_n(X)=[X,U(n)]\) forms a group with the pointwise multiplication. If \(2n>\dim X\), \(U_n(X)\) coincides with \(\widetilde{K^1}(X)\). In [\textit{H. Hamanaka} and \textit{A. Kono}, J. Math. Kyoto Univ. 43, No. 2, 333--348 (2003; Zbl 1070.55007)], the authors showed that, for \(\dim X=2n\), \([X,U(n)]\) is an extension of \(\widetilde{K^1}(X)\) by \(N_n(X)\), where \(N_n(X)\) is a group defined as the cokernel of a map \(\Theta :\widetilde{K^0}(X)\rightarrow H^{2n}(X;\mathbb{Z})\). When \(\widetilde{K^1}(X)\) is a free \(\mathbb{Z}\)-module, the extension \(0\rightarrow N_n(X)\rightarrow U_n(X)\rightarrow \widetilde{K^1}(X)\rightarrow 0\) is determined by the commutators between the inverse images in \(U_n(X)\) of the elements in \(\widetilde{K^1}(X)\). In this paper, the author offers another interpretation of \(N_n(X)\) as the image of a map \(\rho^*:U_n(\bigvee_{i\in I}S^{2n}) \rightarrow U_n(X)\). This image can be determined by the Adams \(e\)-invariants. When \(\widetilde{K^1}(X)\) is not free the above extension is determined by some secondary compositions. As applications, the author determines \([SO(4),U(3)]\), and also shows examples in which \(U_n(X)\) is commutative but the above extension is not trivial.
scientific article; zbMATH DE number 2124480