Subject: Re: question about framed bordism
From: "Dr A. J. Baker"
Date: Thu, 15 Mar 2007 15:00:12 -0000 (GMT)
Dear Johannes,
For principal coverings with fibre a finite group, G say,this is really
about the transfer
\pi^S_nBG_+ -> \pi^S_nEG_+ = \pi^S_n
and this was studied quite a bit for symmetric groups, and cyclic groups
(and more generally the circle).
You might look at the reviews
MR0418091
MR0377873
MR0309109
and other sources I can't think of offhand.
More locally look at
K. Knapp, Some application of ${\rm K}$-theory to framed bordism;
$e$-invariant and transfer, Habilitationsschrift, Bonn, 1979.
Hope that helps,
Andy
>> Subject: A question for the topology list
>> From: Johannes Ebert
>> Date: Wed, 14 Mar 2007 14:04:55 +0100 (CET)
>>
>> here is a question for the toplogy list. You all know that the n th
stable
>> homotopy group of spheres is framed bordism of framed n-manifolds. Now
I
>> consider a closed n- manifold M and a framing a of M. Let f:N \to M be
a
>> finite covering. This induces a framing f^*a of N. Here is the
question:
>>
>> Is there any general statement which relates the elements [M,a] and
>> [N,f^*a] in the stable homotopy group?
>>
>> Best,
>> Johannes
>>
Dr A. J. Baker
Department of Mathematics
University of Glasgow
Glasgow
G12 8QW
Scotland
phone (0141) 330 6140
fax (0141) 330 4111
a.baker@maths.gla.ac.uk
http://www.maths.gla.ac.uk/~ajb/