Formality is preserved under domination

TL;DR

This paper proves that the property of formality in algebraic topology is preserved under certain types of maps between manifolds and Poincaré duality spaces, using $A_ abla$-algebra techniques.

Contribution

It establishes a new criterion showing that formality is maintained under maps from formal spaces to manifolds or Poincaré duality spaces, with a key algebraic technical result.

Findings

Formality is preserved under maps from formal manifolds to certain spaces.

A homotopy $A_ abla$-bimodule retract ensures the transfer of formality.

The main technical result links the formality of $A_ abla$-algebras via bimodule retracts.

Abstract

If a closed orientable manifold (resp. rational Poincar\'e duality space) $X$ receives a map $Y \to X$ from a formal manifold (resp. space) $Y$ that hits a fundamental class, then $X$ is formal. The main technical ingredient in the proof states that given a map of $A_\infty$-algebras $A\to B$ admitting a homotopy $A$-bimodule retract, formality of $B$ implies that of $A$.