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Let be a normal subgroup. Then we have a quotient group and a short exact sequence of groups . This sequence deloops giving the fiber sequence . Is this also a cofiber sequence? I suspect so, but I’m not sure (for sure it is true for abelian groups). Yet the nLab page on cofiber sequences it seems this example is not discussed, so maybe I’m wrong here.
The reason I’m interested in being a cofiber sequence is that this would give a natural description of representations of the quotient as representations of together with the datum of a trivialization of their restriction to (actually proving by hand this equivalence would be a proof that the sequence is a cofiber sequence, but if the result is already known (either as true or as false) I’d avoid working this out in detail myself).
I think it may not be. is a crossed module, and acts by outer automorphisms on , and so at least in the topological classifying case sense, I think is a fibre bundle with fibre and structure group . Or at least, something like that.
This is related somewhat to figuring out what the nonabelian version of a lifting bundle gerbe is.
Is it really true even for abelian groups if you’re talking about spaces rather than spectra?
Well, for central extensions, yes.
Unless I’m misinterpreting the notation, this seems like it would almost never happen just by inspection of cohomology groups. As a fiber sequence the relationship between the cohomology groups is a spectral sequence; in a cofiber sequence you get a long exact sequence.
Gah, I misread the question, so just ignore what I wrote in #2 and #4.
This is generally false. For example consider the central extension of abelian groups
after taking classifying spaces you get
but the cofiber of the first map (which is a double covering map) is only .
I’ve added this counterexample to the page cofiber sequence.
Hi Chris,
Thanks, that’s a beautiful example!
By the way, the natural morphism is homotopy equivalent the canonical inclusion in this case and so it induces in particular an isomorphism on fundamental groups up to . This I guess should be a general phenomenon, i.e., one should have inducing an iso on and in general (the case being obvious, and the being Seifert-van Kampen theorem), am I right?
Yes, that seems right to me.
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