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nLab > Latest Changes: G-CW complex

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2013

    started G-CW complex.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeNov 1st 2013

    I added a bit about G-CW embedding into presheaves on Or(G).

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    moved the section on smooth G-manifolds from “Properties” to “Examples”.

    added pointer to Waner 80, who attributes this class of examples to Matumoto 71

    diff, v13, current

    • CommentRowNumber4.
    • 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

    • Takao Matumoto, Equivariant K-theory and Fredholm operators, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (pdf)

    which is apparently the origin of the concept for equivariance groups being compact Lie groups.

    (A minute ago I also knew the original reference for finite groups, but now I seem to have lost it…)

    diff, v18, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    I have added pointer to

    for the origin of the concept of G-CW complexes over finite groups.

    It’s interesting that Bredon’s proposal to parametrize over the GG-orbit category precedes the proof of the equivariant Whitehead theorem by 4 years, and the proof of Elmendorf’s theorem by 16 years, given that it’s only the combination of these two theorems which justify Bredon’s idea on deeper grounds.

    diff, v22, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    added pointer to:

    diff, v23, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    added pointer to:

    diff, v24, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    added pointer to:

    diff, v26, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    Have expanded/improved the list of attributions for the Prop. (here) that GG-manifolds are GG-CW complexes.

    diff, v26, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2021

    also pointer to:

    • Sören Illman, Section 2 of: 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, v26, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 13th 2021
    • (edited Sep 13th 2021)

    Have added a remark (here) that GG-CW-complexes for finite groups are equivalently CW-complexes with cellular group action.

    Then I have made explicit (here) – for the simple special case of finite groups GG – that binary products in k-spaces preserve GG-CW complex structure.

    I gather this is still true for compact Lie groups GG, “due to” the equivariant triangulation theorem – but what’s an actual proof of this implication?

    diff, v29, current

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