Almost positively curved generalized Eschenburg spaces

TL;DR

We construct infinite families of almost positively curved manifolds in every dimension $4n-1$ with $n\ge 3$, using free $S^1$-quotients of Wilking's metric on $U(n+1)/U(n-1)$ and generalized Eschenburg spaces $\mathcal{E}_{p,q_1,q_2}$. The almost-positivity criterion is reduced to a two-variable degree-4 polynomial $f_{p,q_1,q_2}$, yielding a precise parametric classification that yields infinitely many examples, including strong inhomogeneity for $n\ge 6$. The paper also constructs free $T^2$-quotients with inherited almost positive curvature, proves that all generalized Eschenburg spaces admit quasi-positive curvature (with corrections to type (iv) planes), and establishes infinitely many examples that are not homotopy equivalent to any homogeneous space. Collectively, these results significantly expand the landscape of inhomogeneous examples with strong curvature properties and connect curvature phenomena to explicit algebraic inequalities and cohomological invariants.

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.

Theorems & Definitions (70)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• Proposition 2.5: Kerin
• Proposition 2.6