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)

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)