On finite H-spaces given by sphere extensions of classical groups
The paper studies classical Lie group bundles over spheres, analyzing which ones admit H-structures, which are loop spaces and which are equivalent to classical Lie groups. Let G(n,d) be the group So(n) if \(d=1\), U(n) if \(d=2\) or Sp(n) if \(d=4\). Consider the classical (''last column'') projection \(G(n,d)\to S^{dn-1}\). This is a principal G(n-1,d) bundle. For an integer \(\lambda\) let M(n,d,\(\lambda)\) be the total space of the G(n-1,d) principal bundle pulled back from G(n,d) by a map \(S^{dn-1}\to S^{dn-1}\) of degree \(\lambda\). The author completes the analysis of the M(n,d,\(\lambda)\)'s that were not studied previously and obtains: Theorem A: M(n,1,\(\lambda)\) is an H-space if and only if: (i) n is odd and \(\lambda =\pm 1\), (ii) \(n=2,4,8\), (iii) \(\lambda\) is odd, n is even. Furthermore, in all the cases where M(n,1,\(\lambda)\) is an H-space it is homotopy equivalent to So(n). Theorem B: M(n,d,\(\lambda)\) for \(d=2\) or 4 has the homotopy type of a loop space if and only if \(\lambda\) \(\not\equiv 0 mod p\) for any prime smaller than 1/2 dn. In this case M(n,d,\(\lambda)\) and G(n,d) are p equivalent for every p. The proof uses among others computations in the Steenrod algebra, higher order cohomology operations and BP (Landweber-Novikov) operations.
classical Lie group bundles over spheres
scientific article; zbMATH DE number 3907355
scientific article