[Paper Review] Section problems for configuration spaces of surfaces

Lei Chen|arXiv (Cornell University)|Aug 26, 2017

Homotopy and Cohomology in Algebraic Topology18 references3 citations

TL;DR

This paper classifies continuous sections of configuration space fibrations for ordered and unordered configurations on surfaces, proving that for $\mathbb{R}^2$ ($n>3$) and $S^2$ ($n>4$), sections are either 'adding a point at infinity' or 'adding a point near $x_k$', while no sections exist for surfaces of genus $g>1$ ($n>1$) or $S^2$ when $n=2$. The classification relies on homotopy-theoretic analysis and the canonical reduction system from Thurston's classification of surface diffeomorphisms.

ABSTRACT

In this paper we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of $n$ ordered points on a surface $S$ of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConf$_n(S)$ be the space of ordered $n$-tuple of distinct points in $S$. Let $f_n(S): ext{PConf}_{n+1}(S) o ext{PConf}_n(S)$ be the map given by $f_n(x_0,x_1,\ldots ,x_n):=(x_1,\ldots ,x_n)$. We classify all continuous sections of $f_n$ up to homotopy by proving the following. 1. If $S=\mathbb{R}^2$ and $n>3$, any section of $f_{n}(S)$ is either "adding a point at infinity" or "adding a point near $x_k$". (We define these two terms in Section 2.1; whether we can define "adding a point near $x_k$" or "adding a point at infinity" depends in a delicate way on properties of $S$. ) 2. If $S=S^2$ a $2$-sphere and $n>4$, any section of $f_{n}(S)$ is "adding a point near $x_k$"; if $S=S^2$ and $n=2$, the bundle $f_n(S)$ does not have a section. (We define this term in Section 3.2) 3. If $S=S_g$ a surface of genus $g>1$ and for $n>1$, we give an easy proof that the bundle $f_{n}(S)$ does not have a section.

Motivation & Objective

• To classify all continuous sections of the forgetful fibration $f_n(S): \text{PConf}_{n+1}(S) \to \text{PConf}_n(S)$ for surfaces $S$ of finite type.
• To determine when such sections exist and what their homotopy classes are, particularly distinguishing 'adding a point at infinity' from 'adding a point near $x_k$'.
• To extend the classification to unordered configuration spaces via the quotient fibration $F_n(S)$.
• To resolve the topological obstructions to section existence using fundamental group and cohomology techniques, especially for surfaces of genus $g>1$.

Proposed method

• Use of the canonical reduction system from Thurston's classification of surface diffeomorphisms to analyze monodromy and section obstructions.
• Application of homotopy-theoretic techniques to classify sections up to homotopy, distinguishing between 'adding at infinity' and 'near $x_k$' types.
• Analysis of induced maps on fundamental groups and cohomology, particularly $f_*: \pi_1(\text{PConf}_n(S)) \to \pi_1(S)$ and $f^*: H^1(S) \to H^1(\text{PConf}_n(S))$.
• Use of Poincaré duality and cup product structures to detect nontrivial pullbacks of the diagonal class $[\triangle]$ in $S_g \times S_g$.
• Reduction of the problem to the image of $f_*$ factoring through a forgetful map $p_{i*}$ or being cyclic, followed by contradiction arguments via cohomology.
• Leveraging known results on braid groups and configuration space stability, while introducing new tools via the lantern relation and symplectic bases.

Experimental results

Research questions

• RQ1 When does the fibration $f_n(S): \text{PConf}_{n+1}(S) \to \text{PConf}_n(S)$ admit a continuous section for $S = \mathbb{R}^2$ and $n > 3$?
• RQ2 For $S = S^2$, when do sections exist, and what are their homotopy types for $n=2$, $n>4$, and $n=3$?
• RQ3 Does the fibration $f_n(S_g)$ admit a section for surfaces of genus $g > 1$ and $n > 1$?
• RQ4 How do the sections of the ordered configuration space fibration relate to the unordered case via quotienting by $\Sigma_n$?
• RQ5 What are the homotopy classes of sections for the multi-section fibration $\text{PConf}_{m}(S - \{x_1,\dots,x_n\})/\Sigma_m \to \text{PConf}_{n+m}(S)/\Sigma_m$?

Key findings

• For $S = \mathbb{R}^2$ and $n > 3$, every continuous section of $f_n(S)$ is homotopic to either 'adding a point at infinity' or 'adding a point near $x_k$' for some $1 \leq k \leq n$.
• For $S = S^2$ and $n = 2$, the fibration $f_n(S)$ has no continuous section.
• For $S = S^2$ and $n > 4$, every continuous section of $f_n(S)$ is homotopic to 'adding a point near $x_k$' for some $k$.
• For $S = S_g$ with $g > 1$ and $n > 1$, the fibration $f_n(S)$ admits no continuous section, as shown via contradiction in cohomology and fundamental group analysis.
• In the unordered case, $F_n(\mathbb{R}^2)$ has sections only of type 'adding a point at infinity' for $n > 3$, while $F_n(S^2)$ has no sections for $n = 2,3$ or $n > 4$.
• The proof for genus $g > 1$ relies on showing that any section induces a map $f: \text{PConf}_n(S_g) \to S_g$ whose pullback of the diagonal class leads to a contradiction unless the image of $f_*$ is cyclic or factors through a forgetful map, both of which fail under cohomological constraints.