Subject: Re: two postings
From: Martin Cadek
Date: Fri, 26 Jan 2007 08:40:49 +0100
On Thu, Jan 25, 2007 at 11:39:20AM -0500, Don Davis wrote:
>> Subject: nonorietable bundles
>> From: "John Olsen"
>> Date: Thu, 25 Jan 2007 13:01:50 -0300
>>
>> What can be said about the cohomology H^*(DE, SE; Q) (or integer
>> coefficients) if the vector
>> bundle E over B is nonorientable?
>> Are the dim in which they occur the same as if the bundle was
orientable?
>> Are the dimensions then bounded above by those as if the bundle is
>> orientable?
In this case there is also a Thom class in cohomology with local
coefficients and Thom isomorphism but between cohomology groups with
different local coefficients. For a precise statement see the first
proposition in my paper The cohomology of BO(n) with twisted integer
coefficients, J. Math. Kyoto Univ. 39(1999), 277-286, or Spanier's book,
exercise F in Chapter I and J in Chapter V.
With best wishes, Martin Cadek