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    • 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 G-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 G-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 G-manifolds are G-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 G-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 G – that binary products in k-spaces preserve G-CW complex structure.

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

    diff, v29, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2025

    have spelled out (here) the example of the 2-torus with reflection action on one of the two coordinate axes

    diff, v32, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2025
    • (edited Jul 11th 2025)

    have spelled out (here) the example of the 2-torus with 4-fold rotation action

    diff, v33, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2025

    have spelled out (here) the example of the 2-torus with 3-fold rotation action

    diff, v34, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2025

    have spelled out (here) the example of the 2-torus with 6-fold rotation action

    diff, v35, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2025

    have spelled out (here) the example of the 2-torus with 6-fold dihedral action

    diff, v36, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTime4 days ago

    have spelled out (here) the example of the 2-torus with 2-fold rotation action

    diff, v37, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTime4 days ago

    have spelled out (here) the example of the 2-torus with 3-fold dihedral action

    diff, v38, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTime3 days ago
    • (edited 3 days ago)

    have spelled out (here) the example of the 2-torus with 2-fold dihedral action

    diff, v40, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTime3 days ago

    have spelled out (here) the example of the 2-torus with 4-fold dihedral action

    diff, v41, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTime2 days ago

    have spelled out (here) the example of the 2-torus with 3-fold dihedral action corresponding to the p3m1 wallpaper group (the previous case was p31m).

    diff, v44, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTime2 days ago

    have spelled out (here) the example of the 2-torus with mirror symmetry action corresponding to the cm wallpaper group (the previous case was pm).

    diff, v46, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTime2 days ago

    have spelled out (here) the example of the 2-torus with Dih2 action corresponding to the cmm wallpaper group (the previous case was pmm).

    diff, v49, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTime2 days ago
    • (edited 2 days ago)

    This completes the list of (minimal, I think) G-CW structures for the list of G-actions on the 2-torus corresponding to the 12 non-trivial symmorphic 2D space groups (symmorphic wallpaper groups).

    I have never seen any source spell this out.

    I’ll polish a little more and then will put this material into a separate page in order to !include it here and at wallpaper group.