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nLab > Latest Changes: equivariant Whitehead theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    fixed the pointers to alleged proofs of the equivariant Whitehead theorem.

    What is a canonical citation for this statement, for the unstable case, citable as an actual proof? Is it due to Bredon? Is it in his book?

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018

    Finally I found the kind of citable original citation for the stament that I was after: added pointer to Waner 80, theorem 3.4

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    Made explicit the condition that the equivariance group GG be a compact Lie group.

    (At least that’s what the original proof in Waner 80 assumes. I don’t know if it’s a necessary assumption.)

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    changed the wording regarding Matumoto71: While Waner80 speaks as if Matumoto had a proof of equivariant Whitehead, it seems this refers to Lemma 4.3 in Matumoto, which is only a special case.

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    added pointer to:

    diff, v7, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    added pointer to:

    • {#Kwasik81} Slawomir Kwasik, On the Equivariant Homotopy Type of GG-ANR’s, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 193-194 (2 pages) (jstor:2043921)

    which proves that the G-ANRs considered in James-Segal 78 are of the GG-homotopy type of GG-CW complexes.

    diff, v7, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021
    • (edited Mar 13th 2021)

    added pointer to

    • Takao Matumoto, On GG-CW complexes and a theorem of JHC Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA 18, 363-374, 1971

    but I haven’t actually seen that article yet.

    According to Kwasik 81, the proof of equivariant Whitehead is 5.3 in there.

    In Waner80 the reference to Matumoto is confusing: Both of Matumoto’s articles are listed in the bibliography, but only the other one seems to be referred to in the text, and that other one (at least) does not state the general equivariant Whitehead theorem

    diff, v7, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2021

    added pointer to:

    diff, v8, current

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 16th 2021

    Uploading Matumoto’s paper.

    diff, v10, current

    • CommentRowNumber10.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 16th 2021
    Re #7: I uploaded a PDF of Matumoto's paper.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021
    • (edited Mar 17th 2021)

    Thanks! Am copying your link over also to other entries, such as G-CW complex.

    (By the way, let’s maybe use more informative file names, with maybe the year or parts of the title included in the file name, in case we want to upload more than one article by a given author.)

    It’s curious how Matumoto uses the condition that GG be compact Lie. It appears in Lemma 1.11 as a sufficient condition to imply axiom (f) in his definition 1.2 of GG-cell complex.

    (Generally, I wish the literature on equivariant homotopy theory were more up-front about which exact condition on GG is assumed and for which exact step in the proofs. There is a tiny ratio of discussion this gets over the importance this has for the theory.)

    diff, v12, current

    • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 17th 2021
    > (By the way, let’s maybe use more informative file names, with maybe the year or parts of the title included in the file name, in case we want to upload more than one article by a given author.)

    Yes, that's what I tried to do initially, but the system rejected my file, complaining that the name exceeded 50 characters.

    Can we simply get rid of this limit, or increase it to 300? It must be hard-coded somewhere.
    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2021
    • (edited Mar 18th 2021)

    I guess Richard would know how to change this.

    On the other hand, it makes good sense that filenames fit into one line of a file dialogue box. For uploading articles here I usually make a filename that consists of the author’s name and some abbreviation of the title such that in total (with .pdf-suffix, apparently!) it fits below the 50 character limit.

    I think the point of the filename is only to a) ensure that it’s unique and b) give sufficient hints/mnemonic for what the file might contain. It shouldn’t be too generic (to any bystandanders: don’t upload your thesis as just thesis.pdf!) but also it need not contain full verbatim article titles.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    added pointer to:

    • Sören Illman, Prop. 2.5 in: Equivariant singular homology and cohomology for actions of compact lie groups (doi:10.1007/BFb0070055) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), Proceedings of the Second Conference on Compact Transformation Groups Lecture Notes in Mathematics, vol 298. Springer 1972 (doi:10.1007/BFb0070029)

    diff, v14, current

1 to 14 of 14

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