Free Loop Space Homology of Highly Connected Manifolds

TL;DR

This paper computes the homology of free loop spaces on highly connected manifolds, revealing exponential growth of Betti numbers and providing explicit algebraic structures, using formality, coformality, and Koszul algebra techniques.

Contribution

It introduces explicit formulas for loop space homology and Betti numbers of highly connected manifolds, expanding understanding of their algebraic topology.

Findings

Betti numbers grow exponentially

Explicit formulas for homology and BV-operator

Connection between formality, coformality, and Koszul algebras

Abstract

We calculate the homology of the free loop space of (n-1)-connected closed manifolds of dimension at most 3n-2 (n > 1), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated in earlier work by the first author.