for when the editing functionality is back; to add pointer to this recent review:

- Rita Jimenez Rolland, Jennifer C.H. Wilson,
*Stability properties of moduli spaces*, Notices of the American Mathematical Society 2022 (arXiv:2201.04096)

also the following reference ought to go with Prop. 3.4, as it claims a strengthening in a special case:

- Miguel Xicoténcatl,
*On $\mathbb{Z}_2$-equivariant loop spaces*, Recent developments in algebraic topology, 183—191, Contemp. Math. 407, 2006 (pdf)

just a note for when editing is possible again:

The equivalence in Prop. 3.4 is lacking the symbol for $G$-fixed points on the left.

]]>added pointer to:

- Weiyan Chen,
*Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting*(arXiv:1603.03931)

added pointer to:

- Lukas Brantner, Jeremy Hahn, Ben Knudsen,
*The Lubin-Tate Theory of Configuration Spaces: I*(arXiv:1908.11321)

added pointer to:

Victor Vassiliev,

*Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials*(arXiv:1809.05632)Victor Vassiliev,

*Cohomology of spaces of Hopf equivariant maps of spheres*(arXiv:2102.07157)

added a Properties-section on the Atiyah-Sutcliffe construction (here), essentially copied over from the Idea section which I just wrote at *Atiyah-Sutcliffe conjecture*

added this pointer:

- Lucas Williams,
*Configuration Spaces for the Working Undergraduate*, Rose-Hulman Undergraduate Mathematics Journal: Vol. 21 : Iss. 1 , Article 8. (arXiv:1911.11186, rhumj:vol21/iss1/8)

added pointer to this article today, expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces:

- Quoc P. Ho,
*Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras*(https://arxiv.org/abs/2004.00252)

added publication data for the following, and removed arXiv link (on request of the author):

- Sadok Kallel,
*Spaces of particles on manifolds and Generalized Poincaré Dualities*, The Quarterly Journal of Mathematics, Volume 52, Issue 1, 1 March 2001 (doi:10.1093/qjmath/52.1.45)

added pointer to

- Oscar Randal-Williams, section 10 of:
*Embedded Cobordism Categories and Spaces of Manifolds*, Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 (arXiv:0912.2505)

finally added pointer to

- Christopher Beem, David Ben-Zvi, Mathew Bullimore, Tudor Dimofte, Andrew Neitzke,
*Secondary products in supersymmetric field theory*(arXiv:1809.00009)

I have added a graphics illustrating “$S^1$-labeled” configurations (here)

]]>Added the result of Rourke-Sanderson (here):

Let

$G$ be a finite group,

$V$ an orthogonal $G$-linear representation

$X$ a topological G-space

If $X$ is $G$-connected, in that for all subgroups $H \subset G$ the $H$-fixed point subspace $X^H$ is a connected topological space, then the Cohomotopy charge map

$Conf \big( \mathbb{R}^V, X \big) \underoverset{\simeq}{\;cc\;}{\longrightarrow} \Omega^V \Sigma^V X \phantom{AAA} \text{if X is G-connected}$from the equivariant un-ordered $X$-labeled configuration space of points (Def. \ref{EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}) in the corresponding Euclidean G-space to the based $V$-loop space of the $V$-suspension of $X$, is a weak homotopy equivalence.

If $X$ is not necessarily $G$-connected, then this still holds for the group completion of the configuration space, under disjoint union of configurations

$\Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^V, X \big) \underoverset{\simeq}{\;cc\;}{\longrightarrow} \Omega^{V+1} \Sigma^{V+1} X \,.$ ]]>added pointer to this here:

- Richard Manthorpe, Ulrike Tillmann,
*Tubular configurations: equivariant scanning and splitting*, Journal of the London Mathematical SocietyVolume 90, Issue 3 (arxiv:1307.5669, doi:10.1112/jlms/jdu050)

I have moved the discussion of inclusion maps out of the proposition, and expanded just a little (here).

]]>Sure, precision can be introduced if someone needs the fact that it matters homotopically which component of infinity.

]]>Okay. But if we just say it informally anyway, then I would prefer to just write “…by bringing in a point from infinity”.

]]>I added a phrase about the construction.

]]>Okay, so I removed the word “canonical”, if that’s what you mean. (?)

Of course, somebody should type the actual definition of the maps into the entry, and discuss more details. But I am out of steam for the moment.

]]>I’m objecting to you writing that for any $n$ there’s a canonical map

$Conf_n \big( X \big) \overset{i_n}{\longrightarrow} Conf_{n+1} \big( X \big)$Randall-Williams says on p. 6 that this map depends up to homotopy on the path component of the boundary where the extra point is introduced. So there are canonical maps for each path component of the boundary. Theorems A and B specify the dependency on the point on the boundary, $\mathcal{E}$.

]]>Made explicit Segal73’s Theorem 3 (here) before stating the more general version

]]>He defines the maps in a later section of the article. But I think the construction is classical, must be in Cohen somewhere.

]]>Presumably in #59, one must specify a way of adding a point to the $n$ existing points. I see Randall on p. 6 mentions a $\mathcal{E}$ on the boundary, and says that the class only depends on the component of the boundary.

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