Pairings and monomorphisms of classifying spaces (Q1935834)

Having fixed a monomorphism \(i: H \hookrightarrow G\) between compact Lie groups, the authors are interested in computing \((Bi)^\perp(BK, BG)\), the set of homotopy classes of maps \(\alpha: BK \rightarrow BG\) admitting a ``pairing'' \(BK \times BH \rightarrow BG\) restricting to \(Bi\) on \(BH\) and \(\alpha\) on \(BK\). When \(H\) is a semi-simple subgroup of a connected compact Lie group \(G\) of same rank, this set is trivial when \(K\) is connected. Explicit computations are performed for the inclusions \(i: SU(m) \subset SU(n)\) and \(j: Sp(m) \subset Sp(n)\). Here \[ (Bi)^\perp(BSU(k), BSU(n)) \cong [BSU(k), BSU(n-m)] \] and an analogous statement holds for \(j\). When \(n \geq 3\) and \(i: SO(n) \subset SU(n)\), then \[ (Bi)^\perp(BG, BSU(n)) = 0 \] for any connected compact Lie group \(G\).