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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexam starting something here, to be expanded... <a href="https://ncatlab.org/nlab/revision/twisted+cohomotopy/1">v1</a>, <a href="https://ncatlab.org/nlab/show/twisted+cohomotopy">current</a>

   am starting something here, to be expanded…

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 19th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexam taking the liberty of adding pointer to * [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417)) <a href="https://ncatlab.org/nlab/revision/diff/twisted+cohomotopy/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/twisted+cohomotopy/3">v3</a>, <a href="https://ncatlab.org/nlab/show/twisted+cohomotopy">current</a>

   am taking the liberty of adding pointer to

   • Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane (arXiv:1906.07417)

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 19th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexWell, from a cursory look, there aren't too many other references to choose. Cruikshank's [thesis](http://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape9/PQDD_0030/NQ46823.pdf) 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 (<a href="https://doi.org/10.1016/S0166-8641(02)00183-9">arXiv:10.1016/S0166-8641(02)00183-9</a>) I think that's just the stable version.

   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.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJun 19th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I have added the pointer.

   Thanks. I have added the pointer.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 28th 2019
   • (edited Sep 28th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded statement of one form of what we like to call the "[twisted Pontrjagin-Thom theorem](https://ncatlab.org/nlab/show/twisted+cohomotopy#TwistedPontrjaginThomTheorem)", currently it reads as follows: *** Let 1. $X^n$ be a [[closed manifold]] of [[dimension]] $n$; 1. $1 \leq k \in \mathbb{N}$ a [[positive number|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-homomorphism|J]]-[[twisted Cohomotopy|twisted (n+k)-Cohomotopy]] space of $X^n$, hence the [[space of sections]] of the $(n + k)$-[[spherical fibration]] over $X$ which is [[associated fiber bundle|associated]] via the [[tangent bundle]] by the [[O(n)]]-[[action]] on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$ 1. the [[configuration space of points]] on $X^n$ with labels in $S^k$. *** <a href="https://ncatlab.org/nlab/revision/diff/twisted+cohomotopy/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/twisted+cohomotopy/6">v6</a>, <a href="https://ncatlab.org/nlab/show/twisted+cohomotopy">current</a>

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

   ---

   Let

   1. be a closed manifold of dimension ;

   2. a positive natural number.

   Then the scanning map constitutes a weak homotopy equivalence

   between

   1. the J-twisted (n+k)-Cohomotopy space of , hence the space of sections of the -spherical fibration over which is associated via the tangent bundle by the O(n)-action on

   2. the configuration space of points on with labels in .

   ---

   diff, v6, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 7th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Soren Galatius]], Theorem 6.3 in: _Stable homology of automorphism groups of free groups_ ([arXiv:0610216](https://arxiv.org/abs/math/0610216)) * [[Oscar Randal-Williams]], _Embedded Cobordism Categories and Spaces of Manifolds_, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 ([arXiv:0912.2505](https://arxiv.org/abs/0912.2505)) Also took the liberty of adding pointer to * [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies twisted String structure on M5-branes|Twisted Cohomotopy implies twisted String structure on M5-branes]]_ ([arXiv:2002.11093](https://arxiv.org/abs/2002.11093)) <a href="https://ncatlab.org/nlab/revision/diff/twisted+cohomotopy/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/twisted+cohomotopy/11">v11</a>, <a href="https://ncatlab.org/nlab/show/twisted+cohomotopy">current</a>

   added pointer to

   • Soren Galatius, Theorem 6.3 in: Stable homology of automorphism groups of free groups (arXiv:0610216)

   • Oscar Randal-Williams, Embedded Cobordism Categories and Spaces of Manifolds, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (arXiv:0912.2505)

   Also took the liberty of adding pointer to

   • Domenico Fiorenza (nlab), Hisham Sati (nlab), Urs Schreiber (nlab), Twisted Cohomotopy implies twisted String structure on M5-branes (arXiv:2002.11093)

   diff, v11, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 3rd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe 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](https://ncatlab.org/nlab/show/configuration%20space%20of%20points#LoopSpacesOfSuspensions)). 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. <a href="https://ncatlab.org/nlab/revision/diff/twisted+cohomotopy/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/twisted+cohomotopy/15">v15</a>, <a href="https://ncatlab.org/nlab/show/twisted+cohomotopy">current</a>

   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