Almost positively curved generalized Eschenburg spaces

TL;DR

This paper constructs infinitely many new manifolds with almost positive sectional curvature in dimensions 4n-1 for n≥3, including examples not homotopy equivalent to homogeneous spaces for n≥6.

Contribution

It introduces new examples of almost positively curved manifolds in high dimensions, expanding the known landscape and providing the first infinite family outside homogeneous spaces.

Findings

Existence of infinitely many almost positively curved manifolds in dimensions 4n-1 for n≥3.

Many examples are not homotopy equivalent to homogeneous spaces for n≥6.

First infinite family of such manifolds with this property.

Abstract

In each dimension of the form $4n-1$ with $n\geq 3$, we construct infinitely many new examples of manifolds admitting metrics with positive sectional curvature almost everywhere. In addition, we show that if $n\geq 6$, infinitely many of our examples are not homotopy equivalent to any homogeneous space, providing the first infinite family of such examples.