Rational homotopy type of mapping spaces via cohomology algebras

TL;DR

This paper demonstrates that the rational homotopy type of mapping spaces between certain spaces can be fully determined by their cohomology algebras and the rational homotopy type of the target, revealing new structural insights.

Contribution

It establishes a method to determine the rational homotopy type of mapping spaces using cohomology algebras and identifies conditions for H-structures and homotopy equivalences.

Findings

Rational homotopy type determined by cohomology algebra and target space

Existence of H-structures on mapping space components

Homotopy equivalence characterized by Maurer-Cartan elements connected by automorphisms

Abstract

In this paper, we show that for finite $CW$-complexes $X$ and two-stage space $Y$ (for example $n$-spheres $S^n$, homogeneous spaces and $F_0$-spaces), the rational homotopy type of $\map(X, Y)$ is determined by the cohomology algebra $H^*(X; \Q)$ and the rational homotopy type of $Y$. From this, we deduce the existence of H-structures on a component of the mapping space $\map(X, Y)$, assuming the cohomology algebras of $X$ and $Y$ are isomorphism. Finally, we will show that $\map(X, Y; f)\simeq\map(X, Y; f')$ if the corresponding \emph{Maurer-Cartan elements} are connected by an algebra automorphism of $H^\ast(X, \Q)$.