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Ultrafilter! Waffle! - FLMPotD
February 23rd, 2011
05:32 pm

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FLMPotD
Show that if the homology of a spectrum is free, then its Atiyah-Hirzebruch spectral sequence for any (co)homology theory with torsion-free coefficients degenerates at E_2.

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From:oxeador
Date:February 25th, 2011 07:50 am (UTC)
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Waffle! We told you years ago that you were not allowed to use the letter L anymore! Now turn yourself in and nobody will be hurt.
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From:ultrawaffle
Date:February 25th, 2011 02:46 pm (UTC)
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Hey! This is actually really is a cute little problem if you know basic stable homotopy theory!
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From:oxeador
Date:February 25th, 2011 05:53 pm (UTC)
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Are we gonna have to do this the hard way? You've been warned!
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From:ultrawaffle
Date:February 27th, 2011 08:32 am (UTC)
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No, it really is little. Like, given the appropriate prerequisites (which pretty much any second-year grad student in algebraic topology should have), it's not a superhard or scary problem at all.
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From:ch3cooh
Date:February 26th, 2011 08:10 pm (UTC)
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Hehe, I actually really enjoy your FLMPotD posts. My take: I look up every word in them that I don't know and try to understand what each object is.

In this post:
* homology
* spectrum
* spectral sequence
* Atiyah-Hirzebrauch spectral sequence
* cohomology theory
* homology theory
* torsion-free
* degenerates
* E_2

Aka, I recognized the words: show, that, if, the, of, a, then, its, for, any, with, coefficients, at. And homology, cohomology, and spectral sequence are vaguely in the limit of my vocabulary at this point.

Maybe I'm weird, but I like regularly being confronted with the enormity of what I don't know and may or may not ever figure out. :)
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From:ultrawaffle
Date:February 27th, 2011 07:25 am (UTC)
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Well, in this context "coefficients" doesn't have a meaning in itself, rather "coefficients of a (co)homology theory" has a specific meaning. Also, for what it's worth it's basically impossible to know what "spectral sequence" means without also knowing "degenerates" and "E_2".
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