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nLab > Latest Changes: twisted cohomotopy

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2019

    am starting something here, to be expanded…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2019
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 19th 2019

    Well, from a cursory look, there aren’t too many other references to choose. Cruikshank’s thesis has a relevant Chap. 7, perhaps the source for the entry included in cohomotopy

    • James Cruickshank, Twisted homotopy theory and the geometric equivariant 1-stem, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (arXiv:10.1016/S0166-8641(02)00183-9)

    I think that’s just the stable version.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2019

    Thanks. I have added the pointer.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2019
    • (edited Sep 28th 2019)

    added statement of one form of what we like to call the “twisted Pontrjagin-Thom theorem”, currently it reads as follows:

    Let

    1. X n\mathbb ZX^n be a closed manifold of dimension nn;

    2. 1≤k∈ℕ1 \leq k \in \mathbb{N} a positive natural number.

    Then the scanning map constitutes a weak homotopy equivalence

    Maps /BO(n)(X n,S n def+k triv⫽O(n))a J-twisted Cohomotopy space⟵≃scanning mapConf(X n,S k)aconfiguration spaceof points \underset{ \color{blue} { \phantom{a} \atop \text{ J-twisted Cohomotopy space}} }{ Maps_{{}_{/B O(n)}} \Big( X^n \;,\; S^{ \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}} } \!\sslash\! O(n) \Big) } \underoverset {\simeq} { \color{blue} \text{scanning map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }

    between

    1. the J-twisted (n+k)-Cohomotopy space of X nX^n, hence the space of sections of the (n+k)(n + k)-spherical fibration over XX which is associated via the tangent bundle by the O(n)-action on S n+k=S(ℝ n×ℝ k+1)S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})

    2. the configuration space of points on X nX^n with labels in S kS^k.

    diff, v6, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 7th 2020

    added pointer to

    Also took the liberty of adding pointer to

    diff, v11, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2021

    The section titled “Twisted Pontrjagin-Thom theorem” really talked about the “May-Segal theorem” (i.e. the negative codimension version which gives configuration spaces of points, here).

    I have now instead added a brief pointer to twisted Pontrjagin theorem and gave the previous material a new header “Twisted May-Segal theorem”.

    I should really create an entry May-Segal theorem (as it’s somewhat buried at configuration space of points). But not right now.

    diff, v15, current

1 to 7 of 7

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