17 votes
Accepted

Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics?

The answer is no. This relies on two things: A uniquely geodesic proper metric space is contractible; see here for a proof. $C_n$ is not contractible. Indeed, it has many nontrivial homology groups ...
Andy Putman's user avatar
  • 43.4k
13 votes
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Zero differential in Serre spectral sequence for configuration spaces

I'll write $C_n$ for the configuration space, and $X_n$ for $\mathbb{R}^2$ with $n$ points removed. You are presumably thinking about the spectral sequence $$ E_2^{pq} = H^p(C_{n-1};H^q(X_{n-1})) \...
Neil Strickland's user avatar
10 votes

Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?

Since $C_n(G)$ (for any finite $n$ and finite connected $G$ with at least one cycle or vertex of valence at least $3$) and $\Sigma_g$ (for $g\ge 1$) are both $K(\pi,1)$ spaces homeomorphic to CW ...
Gabriel C. Drummond-Cole's user avatar
9 votes
Accepted

Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?

When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in ...
Dan Petersen's user avatar
  • 39.2k
8 votes
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Very symmetric quadrangle in $\Bbb CP^2$

That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a "regular simplex" in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$. In ...
Dmitri Panov's user avatar
  • 28.7k
8 votes
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The symmetric square of a sphere

That cofiber description in the old short paper of James and other famous folks tells you a lot. The map you have turns out to be adjoint to the standard map $\mathbb RP^{n-1} \rightarrow \Omega^n S^n$...
Nicholas Kuhn's user avatar
8 votes
Accepted

Is there a filtered splitting of product labelling spaces?

The answer to your first question is no. And this can be seen by homology considerations. Note that this equivalence induces an isomorphism of Hopf algebras $$ H_*(C(\mathbb R; X \vee Y \vee (X\...
Nicholas Kuhn's user avatar
8 votes

Integral homology of braid groups as a ring

Using the Hopf fibration you can show that $\Omega^2S^2\simeq\mathbb{Z}\times\Omega^2S^3$. Also $\pi_1(\Omega^2S^3)=\pi_3(S^3)=\mathbb{Z}$, so the Universal Coefficient Theorem gives $[\Omega^2S^3,S^...
Neil Strickland's user avatar
7 votes

Dimension of configuration space of triangulated convex polyhedron

A convex polyhedron can be considered as a flat metric with conic singularities on the sphere. This metric is completely determined (up to a constant factor) by conformal structure of the sphere with $...
Alexandre Eremenko's user avatar
7 votes
Accepted

Mixed Hodge structure on configuration spaces

In fact the conclusion in Gorinov's paper seems to be false, see E. Looijenga, "Torelli group action on the configuration space of a surface", arXiv:2008.10556
Oscar Randal-Williams's user avatar
7 votes

How many unit cylinders can touch a unit ball?

For a javascript rendered 3D model of the path of configurations reported by Ogievetsky and Shlosman in arXiv:1805.09833 see this page on my website. The configurations in this path are composed of ...
Yoav Kallus's user avatar
  • 5,926
6 votes
Accepted

Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects

Too long for a comment but it is essentially a comment: It is easy to see that for a Hausdorff space $X$ the topology on the Ran space coincides with the Vietoris topology and for a non-Hausdorff ...
Taras Banakh's user avatar
  • 40.8k
5 votes
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A piecewise-linear or topological Fulton-MacPherson compactification

In this note I used recent results of Chen and Mann rule out the existence of such a topological compactification in all dimensions $\geq 2$.
skupers's user avatar
  • 7,923
5 votes

fundamental group of configuration spaces of ordered points on open Riemann surfaces

I can at least answer your second question. I'll be a bit brief, but let me know if you need more details. Let $M$ be an oriented manifold and $M^{(r)}$ the configuration space as in your question. ...
Dan Petersen's user avatar
  • 39.2k
5 votes
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Dimension of configuration space of triangulated convex polyhedron

For simplicial polyhedra, one can simply add up the degrees of freedom from placing the vertices in space and substract 7 degrees of freedom for rotation, translation, and scaling to obtain $3v-7$ as ...
Yoav Kallus's user avatar
  • 5,926
5 votes
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How many configurations of tubes are there?

Your proof that $X_n$ is path-connected actually directly shows that $Y_n$ is path-connected. Suppose given $n$ disjoint cylinders in $\mathbb R^3$. Pick a plane which is not parallel to any of the ...
David Wärn's user avatar
4 votes

Cohomology of configuration space of a compact manifold

Shameless promotion ahead. If you are interested in cohomology with base field $\mathbb{R}$, then: For simply connected manifolds without boundary you have my paper and a paper of Campos and ...
Najib Idrissi's user avatar
4 votes
Accepted

cohomology of configuration space of punctured variety

If you read Totaro's paper "Configuration spaces of algebraic varieties" he derives the same cdga calculating the cohomology of $F(X,n)$ as Kriz, but in a different way, via the Leray spectral ...
Dan Petersen's user avatar
  • 39.2k
4 votes

Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?

It has come to my attention that this example has been computed independently (and more directly) by Safia Chettih and Daniel Lütgehetmann. The result is given as Proposition 4.3 in v2 of their ...
3 votes

Is it true, the space of embeddings segments is homotopy equivalent to the subspace of all line segments?

Here are some remarks which may be relevant. First of all it seems to me that the correct topology to use is the Whitney $C^\infty$-topology on the embedding space. Let $M$ be an closed manifold. ...
John Klein's user avatar
  • 18.6k
3 votes
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Fundamental group to groupoid : bijection between homotopy classes?

Just fix a basepoint $a$ and choose a path $u_x$ from $a$ to $x$ for each $x$. Then define $h_{xy}\colon \Pi(a,a)\to\Pi(x,y)$ by $h_{xy}(p)=u_y\circ p\circ u_x^{-1}$. These maps are bijections and ...
Neil Strickland's user avatar
3 votes
Accepted

factorization of the cohomology of configuration space

Here's a geometric construction of a factorization that works for points in $\mathbb R^2$ (and not any other dimension). Given the close relationship between the cohomology rings of configuration ...
Dan Petersen's user avatar
  • 39.2k
3 votes

Cohomology of the moduli space of rational curves with $n$ marked points with spin structure

The cohomology is not mixed Tate for $n\geq 12$, and this possibly can be improved. We can view $\mathcal M_{0,n}$ as the locus of $n$-tuples of points $x_1,\dots,x_n$ in $\mathbb P^1$ which are all ...
Will Sawin's user avatar
  • 135k
3 votes
Accepted

Graded commutativity of the $n$th Browder bracket

Both papers choose a normalization that is different from yours: they define $$ [a,b] = (-1)^{na+1} s(a \otimes b). $$ This is definition 5.7 in Cohen's paper that you mentioned. The reason for this ...
Tyler Lawson's user avatar
  • 51.1k
3 votes
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Configurations of $n$ points modulo isometries of the ambient space

Let us consider the case when $M$ is a hyperbolic plane, $M=\mathbb H^2$ and restrict to orientation preserving isometries of $\mathbb H^2$. Let's identify $\mathbb H^2$ with the open radius $1$ disk ...
Dmitri Panov's user avatar
  • 28.7k
3 votes

Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?

Nice question! Here is a partial answer which was too long for a comment. A necessary condition for collapse at $E_2$ is that the homomorphism $i^*: H^*(\operatorname{PConf}_k(M))\to H^*(\...
Mark Grant's user avatar
2 votes

cohomology of configuration space of punctured variety

The question is addressed by Berceanu, Markl, and Papadima in Multiplicative models for configuration spaces of algebraic varieties, Topology 44 (2005), no. 2, 415–440.
Alex Suciu's user avatar
  • 2,153
2 votes
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Topological Complexity $TC$ of two robots moving on number $8$

The topological complexity of $X$ is the minimum number of open sets needed to cover $X\times X$, on each of which the path fibration $X^I\to X\times X$ admits a local section. If I understood your ...
Mark Grant's user avatar
2 votes

Topological Complexity $TC$ of two robots moving on number $8$

If I understand your question properly, you're asking for the minimum number of open subsets of a wedge of seven circles such that you have unambiguous planning to get from point to point in each ...
Jeff Strom's user avatar
  • 12.5k
2 votes

Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics?

This is just a long comment, and a pretty speculative one at that. However it might perhaps be of interest to you since: there is a natural connection to physics, the construction only works in three ...
Oliver Nash's user avatar
  • 1,404

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