On $[X,U(n)]$ when $\mathrm{dim} X$ is $2n$

Odd primary homotopy types of SU(n)–gauge groups

“…for some map ν. Hamanaka and Kono [5,Proposition 5.2] showed that λ can be chosen so that λ * •(1∧j n ) * is a degree one isomorphism in dimensions 2n and 2n+2. The homotopy commutativity of the left square in (3) therefore implies that ν * is a degree 1 isomorphism in dimensions 2n and 2n + 2.…”

On Samelson products in p-localized unitary groups

“…, ±s n+p−2 ) where s k is the primitive characteristic class in H 2k (BU; Z) satisfying ch k = s k /k!. Here ch k is the universal kth Chern character (see [4]). Thus, we set Θ p : K 0 (X) → p−2 k=0 H 2n+2k (X; Z (p) ) as Θ p = (s n (α), .…”

“…We shall recall the construction of γ in [4]. First we set maps j and k as the following compositions:…”

“…In [4] we constructed maps j : I f → BU(n) and k : C f → BU which are deformations of the compositions…”

“…For a CW-complex X, we call the group of homotopy classes of maps [X, U (n)] the unstable K 1 -theory of X and denote it by U n (X). As seen in [4], U n (X) is equal to K 1 (X), if dim X < 2n, and is a central extension of K 1 (X) if dim X = 2n. Moreover, U n (X) is a second stage central extension of a subgroup of K 1 (X), when dim X = 2n + 1, and its extension is studied in [5,6].…”