Subject: Re: six postings From: "Nicholas J. Kuhn" Date: Fri, 24 Sep 2004 18:54:24 -0400 To: Don Davis CC: njk4x@virginia.edu An answer to Allen Hatcher's question: I think your question fits the following general set up. Let G be a compact Lie group with maximal torus T. Many proofs of the maximal torus theorem also yield the calculation that the Euler characteristic of G/T, e(G/T), is the order of the Weyl group W(G). From this it is pretty easy to deduce that if K is a closed subgroup of G of maximal rank, then e(G/K) = |W(G)|/|W(K)|. Since Grassmanians are such G/K's, one gets the calculation of their Euler characteristics. Nick Kuhn > Subject: question for the list > From: Allen Hatcher > Date: Fri, 24 Sep 2004 14:19:58 -0400 > > Does anyone know a reference to quote for the formula for the Euler > characteristic of the Grassmann manifold of k-planes in n-space, in > particular for the case k = 2 ? > > Remarks: A cell structure is given in Milnor-Stasheff, with a > description of the number of cells in each dimension in terms of > partitions, so it's just a combinatorial question, rather simple for > k = 2. But I don't remember seeing a formula in the literature. > > Allen Hatcher >