  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 $|\langle \beta, Z
\rangle| = \Omega(L^6/\text{log}L)$.
