Subject: Re: two questions Date: 09 Oct 2001 10:53:49 EDT From: Martin.A.Arkowitz@Dartmouth.EDU (Martin A. Arkowitz) To: dmd1@lehigh.edu --- You wrote: Can anyone tell me whether principal Spin(7) bundles over S^15 whose total spaces are H-spaces have been completely determined? Whether the homotopy associative ones have been identified? Whether there is more than one homotopy Spin(9) among the total spaces of principal Spin(7) bundles over S^15? --- end of quote --- Dear Don, Here is a partial response to Dearricott's question. Let E_k be the Spin(7) bundle over S^15 induced from the fibration Spin(7)-->Spin(9)-->S^15 via a map on S^15 of degree k. Then it is known that if k is odd and either 3 does not divide k or 9 divides k, then E_k is an H-space. If k is odd and 3 does not divide k, then E_k is a homotopy associative H-space. These results and similar ones can be found in my notes "Localization and H-spaces", Lecture Notes Series No. 44 (Aarhus University), 1976. --Martin