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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 19th 2019

am starting something here, to be expanded…

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 19th 2019

am taking the liberty of adding pointer to

• 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. $X^n$ be a closed manifold of dimension $n$;

2. $1 \leq k \in \mathbb{N}$ a positive natural number.

Then the scanning map constitutes a weak homotopy equivalence

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

between

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

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

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 7th 2020

Also took the liberty of adding pointer to

• 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.