The homology of the little disks operad

TL;DR

This paper provides an accessible computation of the homology of the little disks operad, demonstrating that the homology of a d-fold loop space forms a Poisson algebra, and introduces new graph-based pairing results.

Contribution

It offers an elementary approach to computing homology of the little disks operad and introduces novel graph and tree pairings in the cohomology and homology pairing.

Findings

Homology of little disks operad is a Poisson algebra.

Identified pairing between homology and cohomology as graphs and trees.

Analyzed cooperad structure on cohomology.

Abstract

In this expository paper we give an elementary, hands-on computation of the homology of the little disks operad, showing that the homology of a $d-fold loop space is a Poisson algebra. One aim is to familiarize a greater audience with Euclidean configuration spaces, using tools accessible to second-year graduate students. We also give a brief introduction to the theory of operads. New results include identifying the pairing between homology and cohomology of these spaces as a pairing of graphs and trees, and treating the cooperad structure on cohomology.