On the homotopy type of the space of Sullivan diagrams

TL;DR

This paper investigates the homotopy type of the harmonic compactification of moduli spaces of 2-cobordisms, revealing connectivity properties, fundamental groups, and non-trivial homology classes, with implications for string topology.

Contribution

It provides new results on the connectivity, fundamental groups, and homology of harmonic compactifications of moduli spaces, extending genus stabilization maps and identifying non-trivial classes.

Findings

Connectivity increases with incoming boundary components.

Fundamental group is non-trivial for certain cobordisms.

Constructs infinite families of non-trivial homology classes.

Abstract

We study the homotopy type of the harmonic compactification of the moduli space of a 2-cobordism S with one outgoing boundary component, or equivalently of the space of Sullivan diagrams of type S on one circle. Our results are of two types: vanishing and non-vanishing. In our vanishing results we are able to show that the connectivity of the harmonic compactification increases with the number of incoming boundary components. Moreover, we extend the genus stabilization maps of moduli spaces to the harmonic compactification and show that the connectivity of these maps increases with the genus and number of incoming boundary components. In our non-vanishing results we compute the non-trivial fundamental group of the harmonic compactification of the cobordism S of any genus with two unenumerated punctures and empty incoming boundary. Moreover, we construct five infinite families of non-trivial homology classes of the harmonic compactification, two of which correspond to non-trivial higher string topology operations.