arXiv subjects

McFeely Jackson Goodman

Publications and source records attributed to McFeely Jackson Goodman.

6 recordsLinked to original sources

arXiv2025

Curvature operators and rational cobordism

We determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted $\hat A$ genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors. These inequalities yield surgery-stable curvature conditions tailored to annihilate further rational cobordism invariants, such as the Witten genus, elliptic genus, signature, and even the rational cobordism class itself.

math.DG↗

arXiv2023

Extremality and rigidity for scalar curvature in dimension four

Following Gromov, a Riemannian manifold is called area-extremal if any modification that increases scalar curvature must decrease the area of some tangent 2-plane. We prove that large classes of compact 4-manifolds, with or without boundary, with nonnegative sectional curvature are area-extremal. We also show that all regions of positive sectional curvature on 4-manifolds are locally area-extremal. These results are obtained analyzing sections in the kernel of a twisted Dirac operator constructed from pairs of metrics, and using the Finsler--Thorpe trick for sectional curvature bounds in dimension 4.

math.DG↗

arXiv2022

Moduli spaces of nonnegatively curved metrics on exotic spheres

We show that the moduli space of nonnegatively curved metrics on each member of a large class of 2-connected 7-manifolds, including each smooth manifold homeomorphic to $S^7$, has infinitely many connected components. The components are distinguished using the Kreck-Stolz $s$-invariant computed for metrics constructed by Goette, Kerin and Shankar. The invariant is computed by extending each metric to the total space of an orbifold disc bundle and applying generalizations of the Atiyah-Patodi-Singer index theorem for orbifolds with boundary. We develop methods for computing characteristic classes and integrals of characteristic forms appearing in index theorems for orbifolds, in particular orbifolds constructed using Lie group actions of cohomogeneity one.

math.DG↗

arXiv2022

The moduli Space of nonnegatively curved metrics on quotients of $S^2\times S^3$ by involutions

We show that for an orientable non-spin manifold with fundamental group $\mathbb{Z}_2$ and universal cover $S^2\times S^3,$ the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The representatives of the components are quotients of the standard metric on $S^3\times S^3$ or metrics on Brieskorn varieties previously constructed using cohomogeneity one actions. The components are distinguished using the relative $η$ invariant of the spin$^c$ Dirac operator computed by means of a Lefschetz fixed point theorem.

math.DG↗

arXiv2020

Moduli spaces of Ricci positive metrics in dimension five

We use the $η$ invariants of spin$^c$ Dirac operators to distinguish connected components of moduli spaces of Riemannian metrics with positive Ricci curvature. We then find infinitely many non-diffeomorphic five dimensional manifolds for which these moduli spaces each have infinitely many components. The manifolds are total spaces of principal $S^1$ bundles over $\#^a\mathbb{C}P^2\#^b\overline{\mathbb{C}P^2}$ and the metrics are lifted from Ricci positive metrics on the bases. Along the way we classify 5-manifolds with fundamental group $\mathbb{Z}_2$ admitting free $S^1$ actions with simply connected quotients.

math.DG↗

arXiv2018

On the moduli spaces of metrics with nonnegative sectional curvature

The Kreck-Stolz s invariant is used to distinguish connected components of the moduli space of positive scalar curvature metrics. We use a formula of Kreck and Stolz to calculate the s invariant for metrics on S^n bundles with nonnegative sectional curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include the first non-homogeneous examples of this type and certain positively curved Eschenburg and Aloff-Wallach spaces.

math.DG↗