Fibrewise homology (Q5890262)

By using the methods of fibrewise homology theory [\textit{M. Crabb} and \textit{I. James}, Fibrewise homotopy theory (1998; Zbl 0905.55001)], the author computes the mod 2 cohomology ring of the classifying space \(B{\mathcal L}SO(n)= {\mathcal L}BSO(n)= \text{map} (S^1,BSO(n))\) of the loop group \({\mathcal L}SO(n)= \text{map} (S^1, SO(n))\), as well as the integral cohomology ring of the space \(\text{map} (S^2, BSU(n))\) of maps from \(S^2\) to \(BSU(n)\). As the author emphasizes, the new method of computation interprets the cohomology rings as duals of fiberwise homology groups admitting an easily described Hopf algebra structure.