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
>