The space of immersed surfaces in a manifold

TL;DR

This paper investigates the rational cohomology of the space of immersed genus g surfaces in a simply-connected manifold, providing stable range computations and insights into torsion phenomena.

Contribution

It offers the first stable range calculations of cohomology for immersed surfaces in manifolds, including torsion information away from (g-1).

Findings

Computed rational cohomology in a stable range

Provided torsion information in cohomology

Extended understanding of immersed surface spaces

Abstract

We study the cohomology of the space of immersed genus g surfaces in a simply-connected manifold. We compute the rational cohomology of this space in a stable range which goes to infinity with g. In fact, in this stable range we are also able to obtain information about torsion in the cohomology of this space, as long as we localise away from (g-1).