The homotopy theory of function spaces: a survey

TL;DR

This survey reviews the homotopy theory of function spaces, including classification, self-equivalences, and free loop spaces, highlighting developments in both classical and localized contexts with algebraic models.

Contribution

It provides a comprehensive overview of recent progress and methods in understanding the homotopy types of function spaces and related structures.

Findings

Progress on classification of homotopy types of function spaces

Development of algebraic models for localized function spaces

Insights into the homotopy theory of aut(X) and LX

Abstract

We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y). We also discuss work on the homotopy theory of the monoid of self-equivalences aut(X) and of the free loop space LX. We consider these topics in both ordinary homotopy theory as well as after localization. In the latter case, we discuss algebraic models for the localization of function spaces and their applications.