added pointer to:

Awais Shaukat, Christian Blanchet,

*Weakly framed surface configurations, Heisenberg homology and Mapping Class Group action*, Archiv der Mathematik**120**(2023) 99–109 [arXiv:2206.11475, doi:10.1007/s00013-022-01793-3]Christian Blanchet,

*Heisenberg homologies of surface configurations*, talk at*QFT and Cobordism*, CQTS (Mar 2023) [web]

Thanks for highlighting this, that’s an interesting note. I’ll record it at *braid group*…

I see that Jon Beardsley has made available notes for a talk, *On Braids and Cobordism Theories*, which discusses the article with Jack Morava I mentioned in #45.

Somehow a program to view $H \mathbb{Z}$ as a Thom spectrum is interpreted in terms of configuration spaces. The notes end with the conjectural

]]>description of integral homology classes as cobordism classes of manifolds with “writhe-free braid orientations.”

added pointer also to:

- Ralph H. Fox, Lee Neuwirth,
*The braid groups*, Math. Scand.**10**(1962) 119-126 $[$doi:10.7146/math.scand.a-10518, pdf, MR150755$]$

added statement (here) that a configuration space of points in the plane is an EM-space

]]>recorded the statement (here) that the forgetful map

$\underset{\{1,\cdots, n+N\}}{Conf}(X) \xrightarrow{\;\;} \underset{\{1,\cdots, N\}}{Conf}(X)$is a Hurewicz fibration

]]>finally remembered #82 and made the fix (in this formula)

This made me also remember #83 and so I added (here) pointer to:

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

This made me also remember #84 and so I added (here) pointer to

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

(I see that AMS finally got the idea to give their *Notices* a web presence. Still no DOI-s for them, though. )

yet earlier occurrence of the configuration space of points in the analysis of particle statistics:

- Michael G. G. Laidlaw, Cécile Morette DeWitt,
*Feynman Functional Integrals for Systems of Indistinguishable Particles*, Phys. Rev. D**3**(1971) 1375 $[$doi:10.1103/PhysRevD.3.1375$]$

added these pointers on early occurrences of configuration spaces of points in physics:

In physics (solid state physics/particle physics) the configuration space of points appears in the discussion of anyon statistics, originally in:

J. M. Leinaas, J. Myrheim, pp. 22 of:

*On the theory of identical particles*,*К теории тождествениых частиц*, Nuovo Cim B 37, 1–23 (1977) (doi:10.1007/BF02727953)Frank Wilczek, p. 959 of:

*Quantum Mechanics of Fractional-Spin Particles*, Phys. Rev. Lett.**49**(1982) 957 (reprinted in Wilczek 1990, p. 166-168) $[$doi:10.1103/PhysRevLett.49.957$]$

added this pointer:

- Martin Palmer,
*Configuration spaces and homological stability*, Oxford University Research Archive (2012) $[$pdf, web$]$

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

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