On \([X, U (n)]\) when \(dim X\) is \(2n+1\)
The paper discusses the group of homotopy classes of maps from \(X\) to \(U(n)\), where \(X\) is a finite complex. When \(dim X = 2n\), the work of the author and \textit{A. Kono} [J. Math. Kyoto Univ. 43, No.~2, 333--348 (2003; Zbl 1070.55007)] describes an exact sequence \[ \tilde K^0(X)\to H^{2n}(X;Z)\to U_n(X)\to \tilde K^1(X)\to 0 \] where \(U_n(X) = [X,U(n)]\). This yields a central extension for \(U_n(X)\). This group is -- in general -- not abelian, and it represents -- in some sense -- an unstable \(\tilde K^1\)-theory. The present paper studies the cases where \(dim X\leq 2n+1\). Here the results are more complicated. In the case where \(n\) is even, the author constructs a secondary operation in order to build the appropriate exact sequence. Nevertheless, the groups \(U_n(X)\) are determined up to central extensions. The methods use Puppe sequences, Postnikov towers, Chern characters, etc. The final section examines the case where \(X\) is a projective space or a stunted projective space. The secondary operation is calculated, yielding specific numerical results.
scientific article; zbMATH DE number 5010268