Algebraic Hopf invariants and rational models for mapping spaces

TL;DR

This paper introduces a new algebraic invariant for maps between finite CW-complexes and rational spaces, proving its completeness and constructing rational models for mapping spaces using $L_{}$- and $C_{}$-structures.

Contribution

It defines a complete invariant $mc_{}(f)$ for maps to rational spaces and constructs algebraic models for mapping spaces using $L_{}$- and $C_{}$-algebras and coalgebras.

Findings

The invariant $mc_{}(f)$ is complete for homotopy classification.

Constructed $L_{}$-models for mapping spaces.

Established algebraic models for rational mapping spaces.

Abstract

In this paper we will define an invariant $mc_{\infty}(f)$ of maps $f:X \rightarrow Y_{\mathbb{Q}}$ between a finite CW-complex and a rational space $Y_{\mathbb{Q}}$. We prove that this invariant is complete, i.e. $mc_{\infty}(f)=mc_{\infty}(g)$ if an only if $f$ and $g$ are homotopic. We will also construct an $L_{\infty}$-model for the based mapping space $Map_*(X,Y_{\mathbb{Q}})$ from a $C_{\infty}$-coalgebra and an $L_{\infty}$-algebra.