On the nonexistence of higher twistings over a point for Borel cohomology of \(K\)-theory
If \(G\) is a compact Lie group and \(X\) is a \(G\)-CW-complex, the twistings of the \(G\)-equivariant \(K\)-theory of the Borel construction \(X \times EG\) are classified by (via the Atiyah-Segal completion theorem and the corresponding non-equivariant twistings) \[ H^1_G(X, \mathbb{Z}/2) \times H^3_G(X, \mathbb{Z}) \times bsu^1_\otimes(X \times_G EG). \] Here \(bsu^1_\otimes(Y) = [Y, BBSU_\otimes]\) for a space \(Y\), and \(\otimes\) denotes the \(H\)-space structure in \(BSU\) given by the tensor product of vector bundles. When \(X\) is a point the group above reduces to \[ H^1(BG, \mathbb{Z}/2) \times H^3(BG, \mathbb{Z}) \times bsu^1_\otimes(BG). \] The author proves that \(bsu^1_\otimes(BG)=0\) for a compact Lie group \(G\) and hence there are no ``higher twistings'' for the Borel cohomology theory associated to \(G\)-equivariant \(K\)-theory, when \(X\) is a point.
scientific article; zbMATH DE number 5689347