Homology of spaces of regular loops in the sphere

TL;DR

This paper computes the singular homology and algebraic structure of the space of immersions of circles into spheres, providing new tools for string topology through enriched Morse spectral sequences.

Contribution

It introduces novel computational methods for homology and algebraic structures of loop spaces in spheres, advancing string topology techniques.

Findings

Computed homology groups of immersion spaces in spheres

Determined the graded commutative algebra structure of these homologies

Developed enriched Morse spectral sequences for loop space fibrations

Abstract

In this paper we compute the singular homology of the space of immersions of the circle into the $n$-sphere. Equipped with Chas-Sullivan's loop product these homology groups are graded commutative algebras, we also compute these algebras. We enrich Morse spectral sequences for fibrations of free loop spaces together with loop products, this offers some new computational tools for string topology.