Subject: Re: question for list From: "John R. Klein" Date: Thu, 23 Sep 2004 10:01:59 -0400 To: Don Davis Dear Yuli, The following arguments show that the only problem might be in dimensions congruent to zero modulo 4. Statement 1: The underlying spherical fibration of the stable tangent bundle is fiber homotopically trivial. Here's an argument: Let M^n be the homology sphere. Choose an embedding M --> S^{n+ k} for k sufficiently large (and k > 2). E be the total space of the spherical normal bundle, and let C be its complement. Then C has the homotopy type of an (k-1)-sphere (by Alexander duality and the hurewicz theorem). Furthermore if j:E --> C is the inclusion and p:E --> M is bundle projection, then E --(p,j)--> M x S^{k-1} is a fiber homotopy trivialization of p. Statement 2: The stable normal (vector) bundle of M is trivial when n is not congruent to 0 mod 4. Here's the argument: step (a) The stable normal bundle corresponds to a class t(M) in [M, BO] if M_0 = M - little disk, then [M_0,BO] is trivial (since BO is a loop space, and suspension(M_0) is contractible). By the Barratt-Puppe sequence, we get [M,BO] = [S^n,BO]. Step (b) There are two cases Case 1: n = 3,5,6,7 mod 8. \pi_n(BO) is trivial in this case. Case 2: n = 1,2 mod 8. then the J-homorphism pi_n(BO) --> pi_n(BG) is injective (by Adams) Don Davis wrote: > From: Yuli Rudyak > Date: Wed, 22 Sep 2004 13:26:28 -0400 (EDT) > > I have a question for the list. It is well-known that homotopy spheres are > stably parallelizable (Kervaire--Milnor). Is the same true for for smooth > HOMOLOGY spheres? What is known about it? >