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nLab > Latest Changes: Universal coefficient theorems

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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 27th 2011

    I was reading Adams’ lectures on generalised cohomology theories and added some stuff from there to universal coefficient theorem about the more general case (including the Kunneth theorem).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2011

    Thanks!

    I have restructured the section outline slightly. Check if you agree. I also made generalized cohomology theories a hyperlink and changed “for the rest of this page” to “for the rest of this section” (since who knows what will be added to the page next!).

    One thing about your material looks like you intended something else: after “The general problems that a Universal Coefficient Theorem should apply to are the following:” there is the line

    • Given E *(X)E_*(X) calculate F *(X)F_*(X).

    First, I think one should say at this point what EE and FF denote. Second, this line repeats four times, verbatim. Probably you meant to change something in each case.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2011

    I have made module-spectrum a redirect to the existing module spectrum, so that the link works now.

    Also I made the word “reduced” point to reduced cohomology.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeOct 27th 2011

    There is also sort of universal coefficient theorem in homology, but it is much weaker and more restrictive. I think there is a bit about it already in Hatcher’s book.

    • CommentRowNumber5.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 27th 2011

    Whoops! That’ll teach me to just cut-and-paste. Yes, those four lines should have been all combinations of super- and sub-script stars. Which means that there’s a universal coefficient theorem for homology, as Zoran says.

    I’ll add in some more information. I’m trying to figure out some conditions when these things hold so as I do that I’ll hopefully understand a bit more what Adams was saying.

    • CommentRowNumber6.
    • CommentAuthorjim_stasheff
    • CommentTimeOct 28th 2011
    @zskoda: There is also sort of universal coefficient theorem in homology, but it is much weaker and more restrictive. I think there is a bit about it already in Hatcher's book.

    Tor is much weaker and restictive than Ext?
    Perhaps you had something else in mind.
    • CommentRowNumber7.
    • CommentAuthorAndrew Stacey
    • CommentTimeOct 31st 2011

    I’ve added a little more to this page, trying to distil some details from Adams’ lecture notes and his blue book about a particular set of circumstances in which this holds.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeOct 31st 2011

    6 Right, Jim, it is not really different :) I was remembering something of the sort I said at the moment when I was writing, what I can not quite reconstruct now.

    • CommentRowNumber9.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 1st 2011

    I just put a little on the page Kunneth theorem to connect it to universal coefficient theorem for generalised cohomology. On that page, there’s a link to a PDF by one Adam Clay. That link no longer works, and I can’t find a suitable replacement. Urs put that link there in the first place, do you remember the document? Is it an important one, or can we just link to any suitable document that contains an exposition of the theorem?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2011

    Hi Andrew,

    thanks. That document was just a random exposition I had handy. Any other one would do fine.

    • CommentRowNumber11.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 1st 2011

    Was it just about the Kunneth theorem for chain complexes (and thus for ordinary (co)homology)? Was there anything pertinent to generalised cohomology?

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2011

    It was just about the simple standard story.

    • CommentRowNumber13.
    • CommentAuthorAndrew Stacey
    • CommentTimeNov 7th 2011

    I’ve added some more for the cases where E *(X)E_*(X) is free or flat. These come from Boardman and Boardman, Johnson, and Wilson’s papers in the Handbook of Algebraic Topology.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2012
    • (edited Aug 28th 2012)

    This here is to the attention of Andrew:

    At universal coefficient theorem where the general idea is introduced, it said:

    Let E *E^* and F *F^* be two generalized cohomology theories and E *E_* and F *F_* two generalized homology theories. Then the general problems that a Universal Coefficient Theorem should apply to are the following:

    1. Given E *(X)E_{*}(X), calculate F *(X)F_{*}(X).

    2. Given E *(X)E_{*}(X), calculate F *(X)F^{*}(X).

    3. Given E *(X)E^{*}(X), calculate F *(X)F_{*}(X).

    4. Given E *(X)E^{*}(X), calculate F *(X)F^{*}(X).

    I have added to the very first sentence the clause

    …such that EE is multiplicative and FF is a module over EE.

    which is the case discussed later on. This or some other relation between EE and FF needs to be stated to make this motivation meaningful.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2012

    I have added the explicit statement of the standard corollary: universal coefficient theorem in topology.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 29th 2012
    • (edited Aug 29th 2012)

    I have written out the direct proof of the UCT in ordinary cohomology (see there), following this note by Boardman.

    • CommentRowNumber17.
    • CommentAuthorAndrew Stacey
    • CommentTimeAug 30th 2012

    Urs, the original is a direct quote from Adams. Maybe I should put it as a quote. He then goes on to say that one should assume some relationship between the two theories, but I thought that point was made in the next paragraph.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2012

    But is it important that we quote the text verbatim? Do you think the addition of the half-sentence

    such that EE is multiplicative and FF is a module over EE.

    is bad? I’d rather have that explanation before that list of four items. The reader is likely to stare at these four items and wonder if he thinks he understands before moving on. And without some previous hint, he won’t be able to understand. The relation between EE and FF is pretty crucial for the whole point.

    • CommentRowNumber19.
    • CommentAuthorAndrew Stacey
    • CommentTimeAug 30th 2012

    True, and I’m not disagreeing with the edit. But as it was originally a quote, I think that if we’re going to change it a little we should change it so that it is definitely different. I’ll ponder it. I suspect I originally had Grand Schemes for working through Adams’ article and working it into the nLab.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 30th 2012
    • (edited Aug 30th 2012)

    I suspect I originally had Grand Schemes for working through Adams’ article and working it into the nLab.

    That would be great, by the way. I am happy that you pointed out that article. This kind of general abstract perspective on the theorem exactly suits my taste :-)

    I don’t have time to work on this myself, though. Instead, for some students I need to spell out more of the nitty-gritty details of the traditional version of the theorem…

    • CommentRowNumber21.
    • CommentAuthorjim_stasheff
    • CommentTimeAug 30th 2012
    @17 the original is a direct quote from Adams.
    or is that Adam?
    or has the `eats, shoots and leaves' gremlin struck again - Adam's?
    • CommentRowNumber22.
    • CommentAuthorAndrew Stacey
    • CommentTimeAug 30th 2012

    I guess the pedantic would write:

    the original is a direct quote from Adams’

    meaning: “from Adams’ lecture notes”.

    The “Adams” in question here is Frank Adams.

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