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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 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 ( 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

    NMM 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 NMM 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 NMM 0N \coloneqq M \setminus M_0 etc. But Prop. 2 assumes that NMM 0N \coloneqq M \setminus M_0 is compact, not that MM is compact. In the examples of interest, while MM is indeed compact, NMM 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 MM 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 MM 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 MM 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 “MM 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

    (MTub(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 nn \in \mathbb{N}, the configuration space of nn points in XX disappearing at the boundary is the topological space

    Conf n(X)((X nΔ 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 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 nΔ 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=* 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)(n𝕟((X nΔ 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 n1,y n1),(x n,y n))((x 1,y 1),,(x n1,y n1))(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 kΔ 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 (n1)(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( d,Y) = C d(Y/Y) = C( d,;Y) Conf n( d) = F nC d(S 0)/F n1C d(S 0) = D n( d,;S 0) Conf n( d,Y) = F nC d(Y/Y)/F n1C d(Y/Y) = D n( 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/YYY* 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. dd \in \mathbb{N}, d1d \geq 1 a natural number with 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( d,Y)nΣ Conf n( 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 d×Y\mathbb{R}^d \times Y vanishing at the boundary and distinct already as points of 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 d×Y\mathbb{R}^d \times Y, vanishing at the boundary and distinct already as points in 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( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))=Ω dΣ d(Y/Y)Σ scanΣ Conf( d,Y)nΣ Conf n( 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 d\mathbb{R}^d, hence with Conf n( d,Y)Conf_n(\mathbb{R}^d, Y) on the right replaced by Conf n( d×Y)Conf_n(\mathbb{R}^d \times Y) in the well-adjusted notation of Def. \ref{ConfigurationSpacesOfnPoints}:

    Maps cp( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))nΣ Conf n( 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
    • CommentTime7 days ago

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

    diff, v32, current

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTime10 hours ago

    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

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