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 ...
13
votes
Accepted
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
$$ E_2^{pq} = H^p(C_{n-1};H^q(X_{n-1})) \...
10
votes
Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?
Since $C_n(G)$ (for any finite $n$ and finite connected $G$ with at least one cycle or vertex of valence at least $3$) and $\Sigma_g$ (for $g\ge 1$) are both $K(\pi,1)$ spaces homeomorphic to CW ...
9
votes
Accepted
Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?
When $n$ is even these spaces are complex algebraic varieties, so the cohomology comes with a mixed Hodge structure. Moreover, this mixed Hodge structure is pure: the cohomology ring is generated in ...
8
votes
Accepted
Very symmetric quadrangle in $\Bbb CP^2$
That is true, since this is so for $\mathbb RP^n$ - take $n+2$ vertices of a "regular simplex" in it -- i.e. take the regular simplex in $S^n$ and project its vertices to $\mathbb RP^n$.
In ...
8
votes
Accepted
The symmetric square of a sphere
That cofiber description in the old short paper of James and other famous folks tells you a lot.
The map you have turns out to be adjoint to the standard map $\mathbb RP^{n-1} \rightarrow \Omega^n S^n$...
8
votes
Accepted
Is there a filtered splitting of product labelling spaces?
The answer to your first question is no. And this can be seen by homology considerations. Note that this equivalence induces an isomorphism of Hopf algebras
$$ H_*(C(\mathbb R; X \vee Y \vee (X\...
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^...
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 $...
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
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 ...
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 ...
5
votes
Accepted
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 $\geq 2$.
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. ...
5
votes
Accepted
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 ...
5
votes
Accepted
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 ...
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 ...
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 ...
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 ...
Community wiki
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. ...
3
votes
Accepted
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 ...
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 ...
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 ...
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 ...
3
votes
Accepted
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 ...
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^*(\...
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.
2
votes
Accepted
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 ...
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 ...
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 ...
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