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    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeOct 28th 2018

    I fixed a link to a pdf file that was giving a general page, and not the file!

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2018

    adding references on configuration spaces of XX appearing as Goodwillie derivatives of Maps(X,−)Maps(X,-)

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 28th 2018

    Added these notes

    diff, v10, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 28th 2018

    have spelled out

    1. the full definition of the configuration space of points with labels in some pointed space

    2. the scanning-map equivalence

    3. the example of such configuration spaces of nn-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…)

    diff, v12, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 28th 2018

    I came across an equivariant version, so have added that

    • Colin Rourke, Brian Sanderson, Equivariant Configuration Spaces, 62(2), October 2000, pp. 544-552,(pdf)

    diff, v13, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018

    Thanks! The scanning map equivalence in the equivariant case would be very useful to have. If you see anything, please let me know.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018

    Oh, sorry, now that I took a real look at the article, I see that this is just what they do. Thanks again.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018

    further polished and expanded the discussion of and around the scanning map equivalence

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018
    • (edited Oct 29th 2018)

    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 ℝ 3\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.)

    diff, v14, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 29th 2018

    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

    • Sadok Kallel, Particle Spaces on Manifolds and Generalized Poincaré Dualities, (arXiv:math/9810067)

    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?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018
    • (edited Oct 29th 2018)

    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.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 29th 2018

    Wow, that is a coincidence.

    I’ll add a paper looking at configurations of extended objects for a geometric model of group completion:

    • Shingo Okuyama, Kazuhisa Shimakawa, Interactions of strings and equivariant homology theories, (arXiv:0903.4667)
    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 29th 2018

    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).

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 29th 2018

    Different campus – NYU Abu Dhabi.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 29th 2018

    Ah, thanks David.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018

    added pointer to

    regarding the relation to graph complexes

    diff, v17, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2018

    Todd, ah, sorry, yes, I am in the land of milk and honey.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2018

    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 (ℝ n) *\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 Ω nS n\Omega^n S^n of negative degree, given that we are wrapping ball-shaped neighbourhoods of all points homeomorphically around S nS^n with degree +1?

    diff, v19, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2018

    Oh, I see. Actually I need to add delooping to make it work. Just a sec…

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 31st 2018

    Where you have

    First, in the special case…

    is AA set equal to S 0S^0 there?

    • CommentRowNumber21.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 31st 2018

    Or perhaps you just missed the AA after Ω nS n\Omega^n S^n.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2018

    Fixed now. Needs more attention, but need to interrupt again..

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 31st 2018
    • (edited Oct 31st 2018)

    I see Segal has Ω nS nA\Omega^n S^n A (though he’s using XX instead of your AA). So his SS is suspension, Σ\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).

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2018
    • (edited Oct 31st 2018)

    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)≃Ω nS nConf_{D^n}(S^0) \simeq \Omega^n S^n, which is not actually the case).

    diff, v20, current

    • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 31st 2018

    Re #18, there’s a description of the degree of an element of what you’re calling Γ˜ XS X TX\widetilde \Gamma_X S_X^{T X} in section 3.1 of Scanning for oriented configuration spaces.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2018
    • (edited Nov 2nd 2018)

    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≔M∖M 0. N \coloneqq M \setminus M_0 \,.

    Now for many of the examples (for instance Examples 2 and 5 and 14 etc. ) this NN is clearly not compact. But Prop. 2 assumes that the NN it is being fed is compact.

    I must be misreading something, but right now I don’t see what.

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 2nd 2018

    Isn’t Prop 1. drawing on Prop. 3, about all NN? Does it also refer to Prop. 2?

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2018
    • (edited Nov 2nd 2018)

    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.

    • CommentRowNumber29.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 2nd 2018
    • (edited Nov 2nd 2018)

    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.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2018

    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.

    • CommentRowNumber31.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 3rd 2018
    • (edited Nov 3rd 2018)

    I think I’m right. Look at example 12 which is working under the assumptions of Prop. 1, including that MM is compact.

    Example 12 has done the work of establishing an equivalence between C(M\M 0,∂M\∂M 0;X)C(M \backslash M_0, \partial M \backslash \partial M_0; X) and map(K,K 0;S mX)map(K, K_0; S^m X) for compact MM. So to establish Prop.1 just requires a stable splitting of C(M\M 0,∂M\∂M 0;X)C(M \backslash M_0, \partial M \backslash \partial M_0; X), and Prop. 3 achieves this.

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    But what I am worried about is that MM being compact doesn’t help if, as you just said yourself, it is N≔M∖M 0N \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)⟶γ⋯ C(N,N_0; X) \overset{\gamma}{\longrightarrow} \cdots

    that Prop. 2 is about, for the case that N≔M∖M 0N \coloneqq M \setminus M_0 etc. But Prop. 2 assumes that N≔M∖M 0N \coloneqq M \setminus M_0 is compact, not that MM is compact. In the examples of interest, while MM is indeed compact, N≔M∖M 0N \coloneqq M \setminus M_0 is not.

    What am I missing?

    • CommentRowNumber33.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 3rd 2018
    • (edited Nov 3rd 2018)

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

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

    where we should replace M 0M_0 by an open tubular neighbourhood to ensure compactness of M∖M 0M \setminus M_0.

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    At the top of p. 182 he has

    where we should replace M 0M_0 by an open tubular neighbourhood to ensure compactness of M∖M 0M \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. )

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    Hm, but then what is “∂M∖M 0\partial M \setminus M_0”? If we agree now that M 0M_0 here has to be replaced by an open tubular neighbourhood Tub(M 0)Tub(M_0), then it now matters whether we read that as

    ∂(M∖Tub(M 0)) \partial\left( M \setminus Tub(M_0) \right)

    or as

    (∂M)∖Tub(M 0) \left( \partial M \right) \setminus Tub(M_0)

    I am guessting we should do the latter. But I wish this were made clearer.

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    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 XX be a manifold, possibly with boundary. For n∈ℕn \in \mathbb{N}, the configuration space of nn points in XX disappearing at the boundary is the topological space

    Conf n(X)≔((X n∖Δ X n)/∂(X n))/Σ(n), \mathrm{Conf}_{n}(X) \;\coloneqq\; \Big( \big( X^n \setminus \mathbf{\Delta}_X^n \big) / \partial(X^n) \Big) /\Sigma(n) \,,

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

    More generally, let YY be another manifold, possibly with boundary. For n∈ℕn \in \mathbb{N}, the configuration space of nn points in X×YX \times Y vanishing at the boundary and distinct as points in XX is the topological space

    Conf n(X,Y)≔(((X n∖Δ X n)×Y n)/Σ(n))/∂(X n×Y n) \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 Σ(n)\Sigma(n) denotes the evident action of the symmetric group by permutation of factors of X×YX \times Y inside X n×Y n≃(X×Y) nX^n \times Y^n \simeq (X \times Y)^n.

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

    Conf n(X)=Conf n(X,*). \mathrm{Conf}_n(X) \;=\; \mathrm{Conf}_n(X,\ast) \,.

    Finally the configuration space of an arbitrary number of points in X×YX \times Y vanishing at the boundary and distinct already as points of XX is the quotient topological space of the disjoint union space

    Conf(X,Y)≔(⊔n∈𝕟((X n∖Δ X n)×Y k)/Σ(n))/∼ 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

    ((x 1,y 1),⋯,(x n−1,y n−1),(x n,y n))∼((x 1,y 1),⋯,(x n−1,y n−1))⇔(x n,y n)∈∂(X×Y). \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 ≤n(X,Y)≔(⊔k∈{1,⋯,n}((X k∖Δ X k)×Y k)/Σ(k))/∼. 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)≃Conf ≤n(X,Y)/Conf ≤(n−1)(X,Y). 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:

    here: Segal 73, Snaith 74: Bödigheimer 87: Conf(ℝ d,Y) = C d(Y/∂Y) = C(ℝ d,∅;Y) Conf n(ℝ d) = F nC d(S 0)/F n−1C d(S 0) = D n(ℝ d,∅;S 0) Conf n(ℝ d,Y) = F nC d(Y/∂Y)/F n−1C d(Y/∂Y) = D n(ℝ d,∅;Y/∂Y) Conf n(X) = D n(X,∂X;S 0) Conf n(X,Y) = D n(X,∂X;Y/∂Y) \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 YY happens to have empty boundary, ∂Y=∅\partial Y = \emptyset, then the pushout

    X/∂Y≔Y⊔∂Y* X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast

    is YY with a disjoint basepoint attached. Notably for Y=*Y =\ast the point space, we have that

    */∂*=S 0 \ast/\partial \ast = S^0

    is the 0-sphere.

    diff, v25, current

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    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

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

    2. YY a manifold, with non-empty boundary so that Y/∂YY / \partial Y is connected,

    there is a stable weak homotopy equivalence

    Σ ∞Conf(ℝ d,Y)⟶≃⊕n∈ℕΣ ∞Conf n(ℝ d,Y) \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

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

    2. the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in ℝ d×Y\mathbb{R}^d \times Y, vanishing at the boundary and distinct already as points in ℝ d\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(ℝ d,Σ d(Y/∂Y))=Maps */(S d,Σ d(Y/∂Y))=Ω dΣ d(Y/∂Y)⟶Σ ∞scan≃Σ ∞Conf(ℝ d,Y)⟶≃⊕n∈ℕΣ ∞Conf n(ℝ d,Y) 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 dd-fold reduced suspension of Y/∂YY / \partial Y.

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

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

    Maps cp(ℝ d,Σ d(Y/∂Y))=Maps */(S d,Σ d(Y/∂Y))⟶≃⊕n∈ℕΣ ∞Conf n(ℝ d×Y) 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)

    diff, v25, current

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2018

    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

    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.

    diff, v26, current

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeNov 9th 2018

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

    diff, v32, current

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2018

    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

    diff, v33, current

    • CommentRowNumber41.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2018

    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

    • CommentRowNumber42.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 5th 2018

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

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

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2018

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

    These are the centers of the skyrmions/calorons.

    • CommentRowNumber44.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 5th 2018

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

    • CommentRowNumber45.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 11th 2019

    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)
    • CommentRowNumber46.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2019

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

    • CommentRowNumber47.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2019
    • (edited Jan 11th 2019)

    All right, so lemma 3.6 in Atiyah-Jones 78 may be read as saying that under the canonical map S 4→BSU(2)S^4 \to B SU(2), Cohomotopy not only sees all instantons on ℝ 4\mathbb{R}^4 via the topological [(ℝ 4) *,S 4] *≃ℤ≃[(ℝ 4) *,BSU(2)] *[(\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.

    • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2019

    added a quick cross-link with D0-D4-brane bound states (here)

    diff, v38, current

    • CommentRowNumber49.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2019

    added statement (here) of the real cohomology ring

    H •(Conf n(ℝ D),ℝ)≃ℝ[{ω ij} i,j∈{1,⋯,n}]/(ω ij=(−1) Dω ji ω ij∧ω ij=0 ω ij∧ω jk+ω jkω ki+ω ki∧ω ij=0fori,j∈{1,⋯,n}) 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)

    diff, v43, current

    • CommentRowNumber50.
    • CommentAuthorUrs
    • CommentTimeOct 5th 2019

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

    diff, v44, current

    • CommentRowNumber51.
    • CommentAuthorUrs
    • CommentTimeOct 6th 2019

    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)

    diff, v46, current

    • CommentRowNumber52.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2019

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

    diff, v49, current

    • CommentRowNumber53.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2019

    and this one:

    diff, v49, current

    • CommentRowNumber54.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2019

    I have introduced more systematic notation for distinguishing between ordered and unordered configurations:

    Now it’s

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

    • Conf n(⋯)Conf_n(\cdots) for un-ordered configurations of nn points.

    I also made the corresponding notational change at graph complex.

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

    diff, v52, current

    • CommentRowNumber55.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 12th 2019

    It occurs to me that just unordered configurations of kk distinct points in an nn-point set could sensibly be denoted (nk)\binom{\mathbf{n}}{\mathbf{k}}, so it could also be suggestive to denote the space of unordered configurations of nn points in a space XX by (Xn)\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 k̲=n(n−1)…(n−k+1)n^\underline{k} = n(n-1)\ldots (n-k+1) to count injections k→n\mathbf{k} \to \mathbf{n}, so one could use X n̲X^\underline{\mathbf{n}} for the space of ordered configurations. Then, in the tradition of categorified algebra,

    (Xn)=X n̲n!.\binom{X}{\mathbf{n}} = \frac{X^\underline{\mathbf{n}}}{\mathbf{n!}}.
    • CommentRowNumber56.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2019

    Thanks, that’s a neat suggestion.

    • CommentRowNumber57.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2019

    added pointer to this:

    • E. Ossa, On the cohomology of configuration spaces, In: Broto C., Carles Casacuberta, Mislin G. (eds.), Algebraic Topology: New Trends in Localization and Periodicity, Progress in Mathematics, vol 136. Birkhäuser Basel (1996) (doi:10.1007/978-3-0348-9018-2_26)

    diff, v60, current

    • CommentRowNumber58.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2019

    added pointer to this:

    • Igor Kriz, On the Rational Homotopy Type of Configuration Spaces, Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 227-237 (jstor:2946581)

    diff, v60, current

    • CommentRowNumber59.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019
    • (edited Oct 18th 2019)

    added the homological stabilization theorem for the unordered configuration spaces (here), from Randall-Williams 13, Theorem A and Threorem B:

    Let XX be

    Then for all n∈ℕn \in \mathbb{N} there are canonical inclusion maps

    Conf n(X)⟶i nConf n+1(X) Conf_n \big( X \big) \overset{i_n}{\longrightarrow} Conf_{n+1} \big( X \big)

    of the unordered configuration soace of nn points in XX (Def. \ref{UnorderedUnlabeledConfigurations}) into that of n+1n + 1 points, such that on ordinary homology with integer coefficients these maps induce split monomorphism in all degrees,

    H •(Conf n(X),ℤ)↪H •(i n,ℤ)H •(Conf n+1(X),ℤ) 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 ≤n/2\leq n/2 these are even isomorphisms

    H p(Conf n(X),ℤ)↪≃H p(i n,ℤ)H p(Conf n+1(X),ℤ)AAAAforp≤n/2. 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 nn:

    H p(Conf n(X),ℚ)↪≃H p(i n,ℚ)H p(Conf n+1(X),ℚ)AAAAforp≤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 \,.

    diff, v61, current

    • CommentRowNumber60.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    Finally added the statement (here) of Theorem 1 in Segal73:

    The Cohomotopy charge map

    Conf(ℝ D)⟶ccΩ DS D Conf \big( \mathbb{R}^D \big) \overset{ cc }{\longrightarrow} \Omega^D S^D

    from the full unordered and unlabeled configuration space (eq:UnorderedUnlabeledConfigurationSpace) of Euclidean space ℝ D\mathbb{R}^D to the DD-fold iterated based loop space of the D-sphere, exhibits the group completion (eq:GroupCompletionOfConfigurationSpaceMonoid) of the configuration space monoid

    ΩB ⊔Conf(ℝ D)⟶≃Ω DS D \Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^D \big) \overset{ \simeq }{\longrightarrow} \Omega^D S^D

    diff, v62, current

    • CommentRowNumber61.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 18th 2019

    Presumably in #59, one must specify a way of adding a point to the nn existing points. I see Randall on p. 6 mentions a ℰ\mathcal{E} on the boundary, and says that the class only depends on the component of the boundary.

    • CommentRowNumber62.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    He defines the maps in a later section of the article. But I think the construction is classical, must be in Cohen somewhere.

    • CommentRowNumber63.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    Made explicit Segal73’s Theorem 3 (here) before stating the more general version

    diff, v64, current

    • CommentRowNumber64.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 18th 2019
    • (edited Oct 18th 2019)

    I’m objecting to you writing that for any nn there’s a canonical map

    Conf n(X)⟶i nConf n+1(X) 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, ℰ\mathcal{E}.

    • CommentRowNumber65.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    Okay, so I removed the word “canonical”, if that’s what you mean. (?)

    Of course, somebody should type the actual definition of the maps into the entry, and discuss more details. But I am out of steam for the moment.

    diff, v65, current

    • CommentRowNumber66.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 18th 2019

    I added a phrase about the construction.

    diff, v66, current

    • CommentRowNumber67.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    Okay. But if we just say it informally anyway, then I would prefer to just write “…by bringing in a point from infinity”.

    • CommentRowNumber68.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 18th 2019

    Sure, precision can be introduced if someone needs the fact that it matters homotopically which component of infinity.

    • CommentRowNumber69.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2019

    I have moved the discussion of inclusion maps out of the proposition, and expanded just a little (here).

    diff, v67, current

    • CommentRowNumber70.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2019

    added pointer to this here:

    diff, v69, current

    • CommentRowNumber71.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2019

    Added the result of Rourke-Sanderson (here):

    Let

    1. GG be a finite group,

    2. VV an orthogonal GG-linear representation

    3. XX a topological G-space

    If XX is GG-connected, in that for all subgroups H⊂GH \subset G the HH-fixed point subspace X HX^H is a connected topological space, then the Cohomotopy charge map

    Conf(ℝ V,X)⟶≃ccΩ VΣ VXAAAif X is G-connected 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 XX-labeled configuration space of points (Def. \ref{EquivariantUnorderedLabeledConfigurationsVanishingWithVanishingLabel}) in the corresponding Euclidean G-space to the based VV-loop space of the VV-suspension of XX, is a weak homotopy equivalence.

    If XX is not necessarily GG-connected, then this still holds for the group completion of the configuration space, under disjoint union of configurations

    ΩB ⊔Conf(ℝ V,X)⟶≃ccΩ V+1Σ V+1X. \Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^V, X \big) \underoverset{\simeq}{\;cc\;}{\longrightarrow} \Omega^{V+1} \Sigma^{V+1} X \,.

    diff, v70, current

    • CommentRowNumber72.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2019

    I have added a graphics illustrating “S 1S^1-labeled” configurations (here)

    diff, v72, current

    • CommentRowNumber73.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2019

    finally added pointer to

    diff, v77, current

    • CommentRowNumber74.
    • CommentAuthorUrs
    • CommentTimeMar 7th 2020

    added pointer to

    diff, v82, current

    • CommentRowNumber75.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2020

    added publication data for the following, and removed arXiv link (on request of the author):

    diff, v83, current

    • CommentRowNumber76.
    • CommentAuthorUrs
    • CommentTimeApr 2nd 2020

    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)

    diff, v84, current

    • CommentRowNumber77.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2021

    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)

    diff, v87, current

    • CommentRowNumber78.
    • CommentAuthorUrs
    • CommentTimeAug 17th 2021

    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

    diff, v92, current

    • CommentRowNumber79.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2021
    • (edited Dec 21st 2021)

    added pointer to:

    diff, v97, current

    • CommentRowNumber80.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2021

    added pointer to:

    diff, v98, current

    • CommentRowNumber81.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2021

    added pointer to:

    • Weiyan Chen, Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting (arXiv:1603.03931)

    diff, v99, current

    • CommentRowNumber82.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2022

    just a note for when editing is possible again:

    The equivalence in Prop. 3.4 is lacking the symbol for GG-fixed points on the left.

    • CommentRowNumber83.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2022

    also the following reference ought to go with Prop. 3.4, as it claims a strengthening in a special case:

    • Miguel Xicoténcatl, On ℤ 2\mathbb{Z}_2-equivariant loop spaces, Recent developments in algebraic topology, 183—191, Contemp. Math. 407, 2006 (pdf)
    • CommentRowNumber84.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2022
    • (edited Jan 13th 2022)

    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)
    • CommentRowNumber85.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2022

    added this pointer:

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

    diff, v101, current

    • CommentRowNumber86.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2022
    • (edited Jun 4th 2022)

    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:

    diff, v103, current

    • CommentRowNumber87.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2022

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

    diff, v105, current

    • CommentRowNumber88.
    • CommentAuthorUrs
    • CommentTimeJun 10th 2022
    • (edited Jun 10th 2022)

    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 ℤ 2\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. )

    diff, v108, current

    • CommentRowNumber89.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022
    • (edited Jun 20th 2022)

    recorded the statement (here) that the forgetful map

    Conf{1,⋯,n+N}(X)→Conf{1,⋯,N}(X) \underset{\{1,\cdots, n+N\}}{Conf}(X) \xrightarrow{\;\;} \underset{\{1,\cdots, N\}}{Conf}(X)

    is a Hurewicz fibration

    diff, v111, current

    • CommentRowNumber90.
    • CommentAuthorUrs
    • CommentTimeJun 20th 2022

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

    diff, v111, current

    • CommentRowNumber91.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2022

    added pointer also to:

    diff, v112, current

    • CommentRowNumber92.
    • CommentAuthorDavid_Corfield
    • CommentTimeFeb 26th 2023

    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 HℤH \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.”

    • CommentRowNumber93.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2023

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

    • CommentRowNumber94.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2023

    added pointer to:

    diff, v118, current

    • CommentRowNumber95.
    • CommentAuthorUrs
    • CommentTimeOct 23rd 2023
    • (edited Oct 23rd 2023)

    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+1n+1-tuple is required to span an nn-dimensional subspace, typically discussed after projective quotienting:

    diff, v123, current

    • CommentRowNumber96.
    • CommentAuthorUrs
    • CommentTimeOct 24th 2023

    added pointer to:

    diff, v124, current

    • CommentRowNumber97.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2023

    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?

    • CommentRowNumber98.
    • CommentAuthorRishi
    • CommentTimeJan 5th 2024

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

    • CommentRowNumber99.
    • CommentAuthorUrs
    • CommentTimeJan 5th 2024

    Thanks for catching this! I have fixed it now.

    diff, v129, current

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