Subject: question for the list
Date: Tue, 10 Jul 2001 18:20:30 -0400
From: Allen Hatcher <hatcher@polygon.math.cornell.edu>

Here is a question about the cohomology of finite-dimensional
Grassmann manifolds.

Consider the complex Grassmann manifold  G_k,n  of  k-planes in  C^n
(or take the real case if you prefer).  Borel showed that the
integral cohomology ring of  G_k,n  is the polynomial ring on the
first  k  Chern classes of the canonical k-plane bundle  E_k,n
modulo certain obvious relations coming from the Whitney sum formula
applied to the sum of  E_k,n  and its orthogonal  (n - k)-plane
bundle. In spite of this simple description, the ring stucture is
well-known to be fairly complicated. My question is whether an
additive basis for the cohomology ring is formed by the monomials of
degree less than or equal to  n - k  in the first  k  Chern classes
of  E_k,n.

As positive evidence, at least the number of these monomials is
correct:  n choose k.  Also the statement seems to be correct in a
few low-dimensional cases I checked by hand.

One can also ask the same question for the K-theory of  G_k,n,  with
the exterior powers of  E_k,n  in place of the Chern classes.

What are some references for these questions? And for that matter,
how about a reference giving an elementary proof of Borel's theorem
(as for example in Stong's cobordism book, but with more details) or
its K-theory analog, which is an exercise in Atiyah's book?

Allen Hatcher