Thanks, that’s a neat suggestion.

]]>It occurs to me that just unordered configurations of $k$ distinct points in an $n$-point set could sensibly be denoted $\binom{\mathbf{n}}{\mathbf{k}}$, so it could also be suggestive to denote the space of unordered configurations of $n$ points in a space $X$ by $\binom{X}{\mathbf{n}}$ (no, I am not saying we should now change the notation of the article).

Similarly, just as one sometimes uses the falling power notation $n^\underline{k} = n(n-1)\ldots (n-k+1)$ to count injections $\mathbf{k} \to \mathbf{n}$, so one could use $X^\underline{\mathbf{n}}$ for the space of ordered configurations. Then, in the tradition of categorified algebra,

$\binom{X}{\mathbf{n}} = \frac{X^\underline{\mathbf{n}}}{\mathbf{n!}}.$ ]]>I have introduced more systematic notation for distinguishing between ordered and unordered configurations:

Now it’s

$\underset{{}^{\{1,\cdots,n \}}}{Conf}(\cdots)$ for ordered configurations of $n$ points

$Conf_n(\cdots)$ for un-ordered configurations of $n$ points.

I also made the corresponding notational change at *graph complex*.

I hope it’s consistent now throughout, otherwise please alert me.

]]>and this one:

- Edward Fadell, Sufian Husseini,
*The space of loops on configuration spaces and the Majer-Terracini index*, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 (euclid:tmna/1476842829)

making a new References-subsection on loop spaces of configuration spaces. Added pointer to

- Fred Cohen, Sam Gitler,
*Loop spaces of configuration spaces, braid-like groups, and knots*, In: Aguadé J., Broto C., Carles Casacuberta (eds.)*Cohomological Methods in Homotopy Theory*. Progress in Mathematics, vol 196. Birkhäuser, Basel (doi:10.1007/978-3-0348-8312-2_7)

have been adding more references on the homology/cohomology of configuration spaces of points:

Yves Félix,

*Rational Betti numbers of configuration spaces*, Topology and its Applications, Volume 102, Issue 2, 8 April 2000, Pages 139-149 (doi:10.1016/S0166-8641(98)00148-5)Thomas Church,

*Homological stability for configuration spaces of manifolds*(arxiv:1602.04748)Christoph Schiessl,

*Betti numbers of unordered configuration spaces of the torus*(arxiv:1602.04748)Christoph Schiessl,

*Integral cohomology of configuration spaces of the sphere*(arxiv:1801.04273)

added statement of the characterization of rational homotopy groups of ordered configuration spaces (here)

]]>added statement (here) of the real cohomology ring

$H^\bullet \Big( Conf_n\big( \mathbb{R}^D \big), \mathbb{R} \Big) \;\simeq\; \mathbb{R}\Big[ \big\{\omega_{i j} \big\}_{i, j \in \{1, \cdots, n\}} \Big] \Big/ \left( \array{ \omega_{i j} = (-1)^D \omega_{j i} \\ \omega_{i j} \wedge \omega_{i j} = 0 \\ \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0 } \;\; \text{for}\; i,j \in \{1, \cdots, n\} \right)$ ]]>added a quick cross-link with *D0-D4-brane bound states* (here)

All right, so lemma 3.6 in Atiyah-Jones 78 may be read as saying that under the canonical map $S^4 \to B SU(2)$, Cohomotopy not only sees all instantons on $\mathbb{R}^4$ via the topological $[(\mathbb{R}^4)^\ast, S^4]_\ast \simeq \mathbb{Z} \simeq [(\mathbb{R}^4)^\ast, B SU(2)]_\ast$, but that this identification is also compatible, via Segal’s “electric field map”, with the standard choices of connections on instantons.

That’s nice. Thanks for highlighting.

]]>Thanks! Hadn’t seen this. Will have a look now.

]]>In case it’s of interest, I came across Beardsley and Morava discussing Segal’s ’electric field map’, p. 11 of

- Jack Morava, Jonathan Beardsley,
*Toward a Galois theory of the integers over the sphere spectrum*, (arXiv:1710.05992)

Did you see Atiyah and Jones relating Segal’s electric field map to instantons in the special case of $\mathbb{R}^4$, from p.104 of

- Topological Aspects of Yang-Mills Theory, (project euclid) ?

What’s playing the role of the charged points in a configuration in your idea?

These are the centers of the skyrmions/calorons.

]]>The original electric field map rather than the scanning map? Placing mild conditions on the electric potential for $k$ charged points in $\mathbb{R}^n$, that amounted to a map from $S^n$ to $S^n$ of degree $k$. Then Segal could say something about when this map from the configuration space to $\Omega^n(S^n)$ induced isomorphisms in homology.

What’s playing the role of the charged points in a configuration in your idea?

]]>This will be cryptic for the time being, but I say it for the heck of it and for the record:

What Segal called the electric field map is not the electric field map… but the pion field map.

As in equation (2) in “Skyrmions from calorons” arXiv:1810.04143

translated via stereographic projection as indicated at representation sphere

]]>added previously missing pointer to page and verse in May 72 for the theorem previously attributed to Segal 73 (Segal gives a quicker re-proof): it’s Theorem 2.7 in May72

]]>added a brief section “Occurrences and Applications” (here) with pointers to developments in other entries.

]]>added pointers to compactifications:

A compactification of configuration spaces of points was introduced in

- William Fulton, Robert MacPherson,
*A compactification of configuration spaces*, Ann. of Math. (2), 139(1):183–225, 1994.

and an operad-structure defined on it (Fulton-MacPherson operad) in

- Ezra Getzler, John Jones,
*Operads, homotopy algebra and iterated integrals for double loop spaces*.

Review includes

- {#LambrechtsVolic14} Pascal Lambrechts, Ismar Volic, section 5 of
*Formality of the little N-disks operad*, Memoirs of the American Mathematical Society ; no. 1079, 2014 (doi:10.1090/memo/1079)

This underlies the models of configuration spaces by graph complexes, see there for more.

]]>added now also the corresponding stament on stable splittings (hence added the same material also at *stable splitting of mapping spaces*). Currently it reads like so:

For

$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,

$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,

there is a stable weak homotopy equivalence

$\Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$between

the suspension spectrum of the configuration space of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. \ref{ConfigurationSpacesOfnPoints})

the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. \ref{ConfigurationSpacesOfnPoints}).

Combined with the stabilization of the scanning map homotopy equivalence from Prop. \ref{ScanningMapEquivalenceOverCartesianSpace} this yields a stable weak homotopy equivalence

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) = \Omega^d \Sigma^d (Y/\partial Y) \underoverset{\Sigma^\infty scan}{\simeq}{\longrightarrow} \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$between the latter direct sum and the suspension spectrum of the mapping space of pointed continuous functions from the d-sphere to the $d$-fold reduced suspension of $Y / \partial Y$.

(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)

In fact by Bödigheimer 87, Example 5 this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)$ ]]>added expanded definition and statement of the equivalence to loop spaces of suspensions from Segal73

also renamed the entry from “configuration space (mathematics)” to “configuration space of points”, which is more informative.

The way I added the Definition now is somewhat ideosyncratic and not as general as what is considered in the literature, but meant to be more suggestive, in particular in the examples that one (not only myself) actually cares about:

Here is how it reads now:

Let $X$ be a manifold, possibly with boundary.
For $n \in \mathbb{N}$, the
* configuration space of $n$ points in $X$ disappearing at the boundary* is the topological space

where $\mathbf{\Delta}_X^n : = \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$ is the fat diagonal in $X^n$ and where $\Sigma(n)$ denotes the evident action of the symmetric group by permutation of factors of $X$ inside $X^n$.

More generally, let $Y$ be another manifold, possibly with boundary.
For $n \in \mathbb{N}$, the
* configuration space of $n$ points in $X \times Y$ vanishing at the boundary and distinct as points in $X$* is the topological space

where now $\Sigma(n)$ denotes the evident action of the symmetric group by permutation of factors of $X \times Y$ inside $X^n \times Y^n \simeq (X \times Y)^n$.

This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:

$\mathrm{Conf}_n(X) \;=\; \mathrm{Conf}_n(X,\ast) \,.$Finally the * configuration space of an arbitrary number of points in $X \times Y$ vanishing at the boundary and distinct already as points of $X$* is the quotient topological space of the disjoint union space

by the equivalence relation $\sim$ given by

$\big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n) \big) \;\sim\; \big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}) \big) \;\;\;\; \Leftrightarrow \;\;\;\; (x_n, y_n) \in \partial (X \times Y) \,.$This is naturally a filtered topological space with filter stages

$Conf_{\leq n}\left( X, Y\right) \;\coloneqq\; \left( \underset{k \in \{1, \cdots, n\}}{\sqcup} \big( ( X^k \setminus \mathbf{\Delta}_X^k ) \times Y^k \big) /\Sigma(k) \right)/\sim \,.$The corresponding quotient topological spaces of the filter stages reproduces the above configuration spaces of a fixed number of points:

$Conf_n(X,Y) \;\simeq\; Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y) \,.$This definition is less general but possibly more suggestive than what is considered in the literature. Concretely, we have the following translations of notation:

$\array{ \text{ here: } && \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: } && \text{ Bödigheimer 87: } \\ \\ Conf(\mathbb{R}^d,Y) &=& C_d( Y/\partial Y ) &=& C( \mathbb{R}^d, \emptyset; Y ) \\ \mathrm{Conf}_n\left( \mathbb{R}^d \right) & = & F_n C_d( S^0 ) / F_{n-1} C_d( S^0 ) & = & D_n\left( \mathbb{R}^d, \emptyset; S^0 \right) \\ \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right) & = & F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y ) & = & D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y \right) \\ \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0 \right) \\ \mathrm{Conf}_n( X, Y ) && &=& D_n\left( X, \partial X; Y/\partial Y \right) }$Notice here that when $Y$ happens to have empty boundary, $\partial Y = \emptyset$, then the pushout

$X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast$is $Y$ with a disjoint basepoint attached. Notably for $Y =\ast$ the point space, we have that

$\ast/\partial \ast = S^0$is the 0-sphere.

]]>Hm, but then what is “$\partial M \setminus M_0$”? If we agree now that $M_0$ here has to be replaced by an open tubular neighbourhood $Tub(M_0)$, then it now matters whether we read that as

$\partial\left( M \setminus Tub(M_0) \right)$or as

$\left( \partial M \right) \setminus Tub(M_0)$I am guessting we should do the latter. But I wish this were made clearer.

]]>At the top of p. 182 he has

where we should replace $M_0$ by an open tubular neighbourhood to ensure compactness of $M \setminus M_0$.

Thanks for catching this side-remark, I had been missing this. That must be what is meant to address the issue!

(Of course one should really trace through the proof to see that this may be done, which I haven’t, but now I am re-assured that at least there wasn’t an evident oversight. )

]]>I see. His own Example 1 would have $M \setminus M_0$ equal to the open interval $(0, 1)$.

What a minute though. At the top of p. 182 he has

]]>where we should replace $M_0$ by an open tubular neighbourhood to ensure compactness of $M \setminus M_0$.