Subject: Some questions From: Johannes Ebert Date: Thu, 24 Nov 2005 18:07:49 +0100 (CET) Hello, I have several questions. First: Consider the groups U_n and SU_n \times U_1. They are diffeomorphic, but not isomorphic as groups. What about the classifying spaces BSU_n \times U_1 and BU_n? Are they homotopy equivalent ot not? I think they are not homotopy equivalent, but I do not know a proof. Second. Thom has given a criterion for realising cohomology classes in a manifold as Poincare dual of classes of submanifolds. Namely, if M is a smooth closed oriented n- manifold and u \in H^k(M;Z), k=2l then there is a n-k dimensional submanifold V \subset M with P.D.([V])=u with (say complex normal bundle) if and only if there is a map f: M \to MU_l, such that the pullback of the universal Thom class via f is u. Now comes the question: Can one further algebraize this condition? Is there a complete answer, for example in terms of Steenrod operations? With this, I mean a statement like this: f as above exists if and only if a certain set of Steenrod operations vanish on u. (Thom only treats low dimensions and other special cases).