added pointer to:

J. Caruso, Stefan Waner:

*An Approximation to $\Omega^n \Sigma^n X$*, Transactions of the American Mathematical Society**265**1 (1981) 147-162 [doi:10.2307/1998487, jstor:1998487, pdf]Shingo Okuyama:

*A simple solution for a group completion problem*, Trends in Mathematics**7**1 (2004) 69-74 [pdf, Okuyama-GroupCompletion.pdf:file]Shingo Okuyama:

*The space of intervals in a Euclidean space*, Algebr. Geom. Topol.**5**(2005) 1555-1572 [arXiv:math/0511645, doi:10.2140/agt.2005.5.1555]

(on enhancing McDuff’s configurations of signed points to signed intervals/cubes in order to get a full model of $\Omega^n \Sigma^n X$ for arbitrary $X$)

]]>I have moved that bibitem out of the subsection “References — As moduli of $Dp/D(p+4)$-branes” into “References — Homology and Cohomology”

]]>pointer

- Stephen Bigelow, Jules Martel.
*Quantum groups from homologies of configuration spaces*(2024). (arXiv:2405.06982).

Added topological complexity of configuration space.

]]>Thanks for catching this! I have fixed it now.

]]>I just wanted to report a suspected typo. I believe in Proposition 3.16, the generators $\omega_{ij}$ should live in $H^{D-1}$, rather than $H^2$

]]>**Question:** Is there any discussion of *spaces of embeddings of normally framed submanifolds*?

I am aware of discussions of “spaces of framed embeddings”, but these are usually about embedding disks into each other, preserving their canonical tangent bundle.

I am after the (spaces of) embeddings of closed submanifolds into closed manifolds, equipped with a trivialization of their normal bundle, so about the structure appearing in Pontrjagin’s theorem, but asking for the spaces these form, not just their cobordism class. Has this been discussed anywhere?

]]>added pointer to:

- Sadok Kallel,
*The Homotopy Type of Graph Configuration Spaces*, talk at CQTS (Oct 2023) [slides:pdf]

added (here) a couple of reverences on spaces of configurations of points “in general position”, where not only any pair of points is required to be non-coincident, but any $n+1$-tuple is required to span an $n$-dimensional subspace, typically discussed after projective quotienting:

Mikhail Kapranov, §2.1 in:

*Chow quotients of Grassmannian I*, Advances in Soviet Mathematics**16**(1993) 29–110 [arXiv:alg-geom/9210002]Nima Arkani-Hamed, Thomas Lam, Marcus Spradlin,

*Positive configuration space*, Commun. Math. Phys.**384**(2021) 909–954 [arXiv:2003.03904, doi:10.1007/s00220-021-04041-x]

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)