Two more responses about orientability, and a posting about genealogy...DMD _______________________________________________________________ Subject: Re: orientability response Date: Thu, 27 Jan 2000 09:32:05 -0500 (EST) From: "Tom Goodwillie,304 Kassar,863-2590,617-926-3565" OOPS- see, I did get it wrong. I said 'even' when I should have said 'odd'. _____________________________________________________________________ From: korbas@dcs.fmph.uniba.sk Subject: Re: orientability response Date: Thu, 27 Jan 2000 16:10:55 +0100 (MET) I rarely have time to react, but now I should say it seems T. Goodwillie is mistaken in case of (real) Grassmann manifolds: the Grassmannian of k-dimensional vector subspaces in Euclidean n-space is in fact non-orientable if and only if n is ODD. A reason: the first Stiefel-Whitney class of the Grassmannian vanishes if and only if n is even (see. e.g. the explicit formulae for the first nine S-W classes of Grassmannians due to V. Bartik and me in Rend. Circ. Mat. Palermo 33, Suppl. 6, 19-29). Other sources on orientability of Grassmannians and more general flag manifolds are e.g. my paper in Annals of Global Analysis and Geometry 3 (1985), 173-184, or my paper (in part a survey) with P. Zvengrowski in Expositiones Math. 12 (1994), 3-30. As to Stiefel manifolds, T. Goodwillie is right. Another reason in this case (for n-k at least 2; the rest is clear): the Stiefel manifold of orthonormal k-frames in Euclidean n-space is (n-k-1)-connected. Still other reason: they are even parallelizable. A good book for generalities on the topology of Grassmann and of Stiefel manifolds is J. Milnor and J. Stasheff's Characteristic Classes. Best regards, Julius Korbas > Date: Wed, 26 Jan 2000 15:41:56 -0500 (EST) > From: "Tom Goodwillie,304 Kassar,863-2590,617-926-3565" > Subject: Re: Orientability question > > > I would like to know what Stiefel and Grassmann manifolds are orientabel > > > and which are not. > > I believe that the Grassmannian of k-planes in n-space is > non-orientable if and only if n is even and 0 > The Stiefel mflds are all orientable. > > This all follows from the fact that if G is a connected Lie group and > H a closed subgroup then G/H is orientable if and only if the (adjoint) > action of H on the vector space g/h is orientation-preserving. > > In the Stiefel cases G=SO(n) and H=SO(k). In the Grassmann cases > G=SO(n) and H is the intersection of G and O(k)xO(n-k). > > - Tom Goodwillie > > P.S. This is the first time I have ever posted anything to a list > in my life. I hope I didn't mget the wrong answer. > >