Date: Thu, 2 Apr 1998 22:26:48 +0100 (BST)
From: luke.hodgkin@kcl.ac.uk (Luke Hodgkin)
Subject: question about arithmetic groups

Dear Don,
Here's a question for your mailing list, which I think quite a few of your
regulars can answer:
I've been working on the homotopy type of BGL(\cal O)+ when \cal O is a
(rather simple kind of) ring of algebraic integers; I want to pass from
2-adic results to local ones. Now Borel gave a description of the real
cohomology, which should give one a hold on the rational homotopy type. For
example, the real cohomology of BSL(\cal O) when \cal O is totally
imaginary looks like that of a product of copies of SU, numbered by the
pairs of embeddings of \cal O in C. However, the argument of Borel's which
I know is all in terms of harmonic forms and such, and doesn't easily
translate into maps, as far as I can see. Has anyone constructed maps (e.g.
rational homotopy equivalences) which induce these cohomology isomorphisms?
The corresponding maps do exist in the 2-adic cases I've been looking at,
but their construction is very etale.
I'd be very grateful for any information - my ignorance is probably great
on the subject.
Luke Hodgkin.