Realization spaces of algebraic structures on cochains

TL;DR

This paper introduces a framework for understanding the space of algebraic structures on chain complexes, using props to include various bialgebras, and provides formulas for classifying and computing their homotopy groups.

Contribution

It develops a general method to compute realization spaces of algebraic structures on chain complexes parametrized by props, including formulas for equivalence classes and higher homotopy groups.

Findings

Computed realization spaces for Poincaré duality on manifolds.

Derived formulas for classifying algebraic structures up to homotopy.

Determined higher homotopy groups via deformation complex cohomology.

Abstract

Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are parametrised by props and thus include various kinds of bialgebras. We give a general formula to compute subsets of equivalences classes of realizations as quotients of automorphism groups, and determine the higher homotopy groups via the cohomology of deformation complexes. As a motivating example, we compute subsets of equivalences classes of realizations of Poincar\'e duality for several examples of manifolds.