On manifolds satisfying stable systolic inequalities

TL;DR

This paper characterizes when stable systolic inequalities hold for manifolds, linking them to cohomology classes and showing they depend on intersection forms, with differences between orientable and nonorientable cases.

Contribution

It establishes a precise criterion for stable systolic inequalities based on cohomology generation and relates the stable systolic constant to intersection forms and fundamental class images.

Findings

Stable systolic inequalities hold iff certain cohomology classes generate top-degree cohomology.

In nonorientable manifolds, such bounds do not exist for stable systoles of dimension ≥ 2.

Stable systolic constants depend only on the image of the fundamental class in Eilenberg-Mac Lane spaces.

Abstract

We show that for closed orientable manifolds the $k$-dimensional stable systole admits a metric-independent volume bound if and only if there are cohomology classes of degree $k$ that generate cohomology in top-degree. Moreover, it turns out that in the nonorientable case such a bound does not exist for stable systoles of dimension at least two. Additionally, we prove that the stable systolic constant depends only on the image of the fundamental class in a suitable Eilenberg-Mac Lane space. Consequently, the stable $k$-systolic constant is completely determined by the multilinear intersection form on $k$-dimensional cohomology.