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I fixed a link to a pdf file that was giving a general page, and not the file!
Added these notes
have spelled out
the full definition of the configuration space of points with labels in some pointed space
the scanning-map equivalence
the example of such configuration spaces of $n$-spheres.
(Our above statement about the sphere spectrum and Ran-spaces is still not right. But there is some relation. Vincent will have more on this…)
Thanks! The scanning map equivalence in the equivariant case would be very useful to have. If you see anything, please let me know.
Oh, sorry, now that I took a real look at the article, I see that this is just what they do. Thanks again.
added more pointers to the original article Segal 73, in particular I added a pointer to Segal’s original description of the “scanning map” as the map that sends a configuration of charged particles in $\mathbb{R}^3$ to the electric field that they generate.
(Though I suspect that this picture is a red herring. It’s interesting though that Segal73 would prefer this over the natural homotopy theoretic picture.)
I get the impression that its being called a ’scanning’ map, because one is scanning the manifold to see what appears of the points in small volumes within the manifold. Is that on the right lines? See bottom of p. 2 of
He says that the scanning map is “closely associated” to the electric field map pp. 21-22. Is there something to say about the latter map in more general situations?
I don’t know. Sadok Kallel was here at NYU AD a few days back and talked about this. Alas, I wasn’t up to speed with configuration spaces yet, which is a bit of a cosmic anti-coincidence now. But I will visit him and his group in Sharjah in February.
Wow, that is a coincidence.
I’ll add a paper looking at configurations of extended objects for a geometric model of group completion:
Urs, it sounds like you’re in the US right now. if you’re going to be in the area again in February, for example at Yale or NYC, please let me know (I live in Connecticut).
Different campus – NYU Abu Dhabi.
Ah, thanks David.
added pointer to
regarding the relation to graph complexes
Todd, ah, sorry, yes, I am in the land of milk and honey.
I have made explicit the “scanning map” (here) in its simple-looking homotopy theoretic form (as opposed to via electric fields) for the special case where the base space is $\left( \mathbb{R}^n\right)^\ast$, as given in section 3 of Segal 73.
Simple as it is, I need to think: How does this map hit an element in $\Omega^n S^n$ of negative degree, given that we are wrapping ball-shaped neighbourhoods of all points homeomorphically around $S^n$ with degree +1?
Oh, I see. Actually I need to add delooping to make it work. Just a sec…
Where you have
First, in the special case…
is $A$ set equal to $S^0$ there?
Or perhaps you just missed the $A$ after $\Omega^n S^n$.
Fixed now. Needs more attention, but need to interrupt again..
I see Segal has $\Omega^n S^n A$ (though he’s using $X$ instead of your $A$). So his $S$ is suspension, $\Sigma$, presumably there, allowing his to play with, SS^n$ as the n-sphere and as the n-th suspension (see Theorem 2 - n-fold reduced suspension).
There was a real mistake in the way I had stated the scanning map equivalence theorem. For the moment I have removed all the related material. Before re-including a fixed version, I will produce a clean one offline now, not to make a mess.
After going through thinking, in turn, that there must be a typo in a) Segal73, b) McDuff75, c) Boedigheimer86, it finally dawned on me, with much help from Vincent, that and how all these are consistent with themselves and with each other (unwinding the fine print in the notation and conventions is a bit of a task here…), but I dare say that there is a mistake in the statement of the theorem in Francis’ lecture (the statement there seems to specialize to $Conf_{D^n}(S^0) \simeq \Omega^n S^n$, which is not actually the case).
Re #18, there’s a description of the degree of an element of what you’re calling $\widetilde \Gamma_X S_X^{T X}$ in section 3.1 of Scanning for oriented configuration spaces.
Am still compiling revised notes offline, as per #24 above.
I feel stuck with the following step in the proof in Boedigheimer 87:
The proof of Prop. 1 (on p. 184) seems to want to use Prop. 2 (on p. 180) with
$N \coloneqq M \setminus M_0 \,.$Now for many of the examples (for instance Examples 2 and 5 and 14 etc. ) this $N$ is clearly not compact. But Prop. 2 assumes that the $N$ it is being fed is compact.
I must be misreading something, but right now I don’t see what.
Isn’t Prop 1. drawing on Prop. 3, about all $N$? Does it also refer to Prop. 2?
I read the proof of Prop 1 there as concatenating Prop 2 with Prop 3: Prop. 2 identifies the mappibg space with the infinite configuration space, Prop. 3 identifies the latter stably with the wedge sum of finite configuration spaces.
But I must be missing something so if you understand it differently, let me know.
The logic is very hard to follow. Where in the proof of Prop 1 he claims to continue examples 7 and 11, does he not mean examples 8 and 12? Example 12 uses the presuppositions of Prop 1, and appeals to Prop 2.
I was also wondering about the relevance and numbering of the Examples being mentioned. But combining Prop. 2 with Prop. 3 as I said does immediately yield a proof – if their assumptions are met – and I read the “now follows from that of” as suggesting just that.
Thanks for looking into it. I have sent an email to the author.
I think I’m right. Look at example 12 which is working under the assumptions of Prop. 1, including that $M$ is compact.
Example 12 has done the work of establishing an equivalence between $C(M \backslash M_0, \partial M \backslash \partial M_0; X)$ and $map(K, K_0; S^m X)$ for compact $M$. So to establish Prop.1 just requires a stable splitting of $C(M \backslash M_0, \partial M \backslash \partial M_0; X)$, and Prop. 3 achieves this.
But what I am worried about is that $M$ being compact doesn’t help if, as you just said yourself, it is $N \coloneqq M \setminus M_0$ that is being fed into Prop. 2.
No?
As you just said, we want to consider the map
$C(N,N_0; X) \overset{\gamma}{\longrightarrow} \cdots$that Prop. 2 is about, for the case that $N \coloneqq M \setminus M_0$ etc. But Prop. 2 assumes that $N \coloneqq M \setminus M_0$ is compact, not that $M$ is compact. In the examples of interest, while $M$ is indeed compact, $N \coloneqq M \setminus M_0$ is not.
What am I missing?
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 $M \setminus M_0$.
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. )
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.
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
$\mathrm{Conf}_{n}(X) \;\coloneqq\; \Big( \big( X^n \setminus \mathbf{\Delta}_X^n \big) / \partial(X^n) \Big) /\Sigma(n) \,,$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
$\mathrm{Conf}_{n}(X,Y) \;\coloneqq\; \Big( \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^n \big) /\Sigma(n) \Big) / \partial(X^n \times Y^n)$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
$Conf\left( X, Y\right) \;\coloneqq\; \left( \underset{n \in \mathbb{n}}{\sqcup} \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^k \big) /\Sigma(n) \right)/\sim$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.
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 pointers to compactifications:
A compactification of configuration spaces of points was introduced in
and an operad-structure defined on it (Fulton-MacPherson operad) in
Review includes
This underlies the models of configuration spaces by graph complexes, see there for more.
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
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?
What’s playing the role of the charged points in a configuration in your idea?
These are the centers of the skyrmions/calorons.
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
In case it’s of interest, I came across Beardsley and Morava discussing Segal’s ’electric field map’, p. 11 of
Thanks! Hadn’t seen this. Will have a look now.
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.
added a quick cross-link with D0-D4-brane bound states (here)
added statement (here) of the real cohomology ring
$H^\bullet \Big( Conf_n\big( \mathbb{R}^D \big), \mathbb{R} \Big) \;\simeq\; \mathbb{R}\Big[ \big\{\omega_{i j} \big\}_{i, j \in \{1, \cdots, n\}} \Big] \Big/ \left( \array{ \omega_{i j} = (-1)^D \omega_{j i} \\ \omega_{i j} \wedge \omega_{i j} = 0 \\ \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0 } \;\; \text{for}\; i,j \in \{1, \cdots, n\} \right)$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)
making a new References-subsection on loop spaces of configuration spaces. Added pointer to
and this one:
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.
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,
$\binom{X}{\mathbf{n}} = \frac{X^\underline{\mathbf{n}}}{\mathbf{n!}}.$Thanks, that’s a neat suggestion.
added pointer to this:
added pointer to this:
added the homological stabilization theorem for the unordered configuration spaces (here), from Randall-Williams 13, Theorem A and Threorem B:
Let $X$ be
which is the interior of a compact manifold with boundary
of dimension $dim(X) \geq 2$.
Then for all $n \in \mathbb{N}$ there are canonical inclusion maps
$Conf_n \big( X \big) \overset{i_n}{\longrightarrow} Conf_{n+1} \big( X \big)$of the unordered configuration soace of $n$ points in $X$ (Def. \ref{UnorderedUnlabeledConfigurations}) into that of $n + 1$ points, such that on ordinary homology with integer coefficients these maps induce split monomorphism in all degrees,
$H_\bullet \big( Conf_n(X) , \mathbb{Z} \big) \overset{ H_\bullet( i_n, \mathbb{Z} ) }{\hookrightarrow} H_\bullet \big( Conf_{n+1}(X) , \mathbb{Z} \big)$and in degrees $\leq n/2$ these are even isomorphisms
$H_p \big( Conf_n(X) , \mathbb{Z} \big) \underoverset{\simeq}{ H_p( i_n, \mathbb{Z} ) }{\hookrightarrow} H_p \big( Conf_{n+1}(X) , \mathbb{Z} \big) \phantom{AAAA} \text{for} \; p \leq n/2 \,.$Finally, for ordinary homology with rational coefficients, these maps induce isomorphisms all the way up to degree $n$:
$H_p \big( Conf_n(X) , \mathbb{Q} \big) \underoverset{\simeq}{ H_p( i_n, \mathbb{Q} ) }{\hookrightarrow} H_p \big( Conf_{n+1}(X) , \mathbb{Q} \big) \phantom{AAAA} \text{for} \; p \leq n \,.$Finally added the statement (here) of Theorem 1 in Segal73:
from the full unordered and unlabeled configuration space (eq:UnorderedUnlabeledConfigurationSpace) of Euclidean space $\mathbb{R}^D$ to the $D$-fold iterated based loop space of the D-sphere, exhibits the group completion (eq:GroupCompletionOfConfigurationSpaceMonoid) of the configuration space monoid
$\Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^D \big) \overset{ \simeq }{\longrightarrow} \Omega^D S^D$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.
He defines the maps in a later section of the article. But I think the construction is classical, must be in Cohen somewhere.
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, $\mathcal{E}$.
Okay. But if we just say it informally anyway, then I would prefer to just write “…by bringing in a point from infinity”.
Sure, precision can be introduced if someone needs the fact that it matters homotopically which component of infinity.
added pointer to this 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 \,.$1 to 72 of 72