Efficient Cycles in Loop Space

TL;DR

This paper explores how the geometry of cycles in the loop space of a Riemannian manifold influences their topology, providing bounds on Gromov's distortion and constructing cycles with significant pairing values.

Contribution

It establishes new bounds relating loop space geometry to topological distortion and constructs cycles with large pairings, improving understanding of Gromov's distortion.

Findings

Upper bounds on pairings imply bounds on Gromov's distortion.

Existence of cycles with pairings of order L^6/log L.

Detection of homotopy classes in complex projective space.

Abstract

This paper investigates how the geometry of a cycle in the loop space of a Riemannian manifold controls its topology. For fixed $\beta \in H^n(\Omega X; \mathbb{R})$ one can ask how large $|\langle \beta, Z \rangle|$ can be for cycles $Z$ supported in loops of length $\leq L$ and of volume $\leq L^{n-1}$ for a suitably defined notion of volume of in loop space. We show that an upper bound to this question provides upper bounds Gromov's distortion of higher homotopy groups. We also show that we can exhibit better lower bounds than are currently known for the corresponding questions for Gromov's distortion. Specifically, we show there exists a $\beta$ detecting the homotopy class of the puncture in $[(\mathbb{CP}^2)^{\#4} \times S^2]^\circ$ and a family of cycles $Z_L$ with the geometric bounds above such that $|\langle \beta, Z \rangle| = \Omega(L^6/\text{log}L)$.