On \([X,U(n)]\) when \(\dim X\) is \(2n\)
For a CW complex \(X\) and the topological group \(U(n)\) the symbol \(U_n(X)\) denotes the group \([X,U(n)]\) of homotopy classes of maps of \(X\) into \(U(n)\). If dim\(X < 2n\), then \(U_n(X)\) equals \(\tilde{K}^1(X)\) and one may say that \(U_n(X)\) is ``the unstable \(\tilde{K}^1\)-theory''. The authors investigate, and obtain several results, the case when \(dim X = 2n\), in which case \(U_n(X)\) may be non-abelian. If \(\dim X \leq 2n\), they find a homomorphism \(\Theta: \tilde{K}^0(X) \to H^{2n}(X;\mathbb Z)\), such that the following exact sequence holds\newline \[ \tilde{K}^0(X) \to H^{2n}(X;\mathbb Z) \to U_n(X) \to \tilde{K}^1(X) \to 0 \] and it is natural. These results are based on investigations of the Coker\(\Theta = N_n(X)\). Among the results obtained we mention the following that has a simple formulation: If in addition \(X\) is finite, then \(N_n(X)\) is a finite Abelian group and the order of any element in \(N_n(X)\) divides \(n!\).
scientific article; zbMATH DE number 2124040