Rational model of the configuration space of two points in a simply connected closed manifold

TL;DR

This paper investigates the rational homotopy type of the configuration space of two points in simply connected closed manifolds, revealing dependence on the manifold's type and constructing algebraic models for odd-dimensional cases.

Contribution

It provides a rational homotopy model for the configuration space of two points in simply connected closed manifolds, distinguishing even and odd dimensions, and introduces a new algebraic construction for odd dimensions.

Findings

For even-dimensional manifolds, the homotopy type depends only on the manifold's rational homotopy type.

For odd-dimensional manifolds, a family of CDGA models $C(x)$ encodes the homotopy type.

In some cases, the model simplifies to $x=0$, capturing the homotopy type completely.

Abstract

Let $M$ be a simply connected closed manifold of dimension $n$. We study the rational homotopy type of the configuration space of 2 points in $M$, $F(M,2)$. When $M$ is even dimensional, we prove that the rational homotopy type of $F(M,2)$ depends only on the rational homotopy type of $M$. When the dimension of $M$ is odd, for every $x\in H^{n-2} (M, \mathbb{Q})$, we construct a commutative differential graded algebra $C(x)$. We prove that for some $x \in H^{n-2} (M; \mathbb{Q})$, $C(x)$ encodes completely the rational homotopy type of $F(M,2)$. For some class of manifolds, we show that we can take $x=0$.