A rational splitting of a based mapping space

TL;DR

This paper establishes conditions under which the rational homotopy type of a based mapping space decomposes into a product of loop spaces, based on the bracket length of attaching maps and the Whitehead length of the target space.

Contribution

It introduces a criterion involving bracket length and Whitehead length that determines when a mapping space's rational homotopy type splits as a product of loop spaces.

Findings

Mapping space decomposes as a product when bracket length exceeds Whitehead length.

Rational homotopy type of complex mapping spaces can be simplified under certain conditions.

Provides a model for analyzing based mapping spaces with CW complex attachments.

Abstract

Let F_*(X, Y) be the space of base-point-preserving maps from a connected finite CW complex X to a connected space Y. Consider a CW complex of the form X cup_{alpha}e^{k+1} and a space Y whose connectivity exceeds the dimension of the adjunction space. Using a Quillen-Sullivan mixed type model for a based mapping space, we prove that, if the bracket length of the attaching map alpha: S^k --> X is greater than the Whitehead length WL(Y) of Y, then F_*(X cup_{alpha}e^{k+1}, Y) has the rational homotopy type of the product space F_*(X, Y) times Omega^{k+1}Y. This result yields that if the bracket lengths of all the attaching maps constructing a finite CW complex X are greater than WL(Y) and the connectivity of Y is greater than or equal to dim X, then the mapping space F_*(X, Y) can be decomposed rationally as the product of iterated loop spaces.