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Christine Escher

Publications and source records attributed to Christine Escher.

9 recordsLinked to original sources
arXiv2025

Computing homology of $\mathbb{Z}_k$-complexes from their quotients

In this paper, we investigate the question of how one can recover the homology of a simplicial complex $X$ equipped with a regular action of a finite group $G$ from the structure of its quotient space $X/G.$ Specifically, we describe a process for enriching the structure of the chain complex $C_\ast(X/G; \mathbb{F})$ using the data of a complex of groups, a framework developed by Bridson and Corsen for encoding the local structure of a group action. We interpret this data through the lens of matrix representations of the acting group, and combine this structure with the standard simplicial boundary matrices for $X/G$ to construct a surrogate chain complex. In the case $G = \mathbb{Z}_k,$ the group ring $\mathbb{F}G$ is commutative and matrices over $\mathbb{F}G$ admit a Smith normal form, allowing us to recover the homology of $G$ from this surrogate complex. This algebraic approach complements the geometric compression algorithm for equivariant simplicial complexes described by Carbone, Nanda, and Naqvi.

math.AT↗
arXiv2024

On Fixed-Point Sets of $\Z_2$-Tori in Positive Curvature

In recent work of Kennard, Khalili Samani, and the last author, they generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on positively curved manifolds to $\mathbb{Z}_2$-tori with a fixed point. They show that if the rank is approximately one-fourth of the dimension of the manifold, then fixed point set components of small co-rank subgroups of the $\Z_2$-torus are homotopy equivalent to spheres, real projective spaces, complex projective spaces, or lens spaces. In this paper, we lower the bound on the rank of the $\mathbb{Z}_2$-torus to approximately $n/6$ and $n/8$ and are able to classify either the integral cohomology ring or the $\mathbb{Z}_2$-cohomology ring, respectively, of the fixed point set of the $\mathbb{Z}_2$-torus.

math.DG↗
arXiv2023

Almost Isotropy-Maximal Manifolds of Non-negative Curvature

We extend the equivariant classification results of Escher and Searle for closed, simply connected, non-negatively curved Riemannian $n$-manifolds admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove that such manifolds are equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to three.

math.DG↗
arXiv2020

Non-negative curvature and torus actions

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres of dimensions greater than or equal to three. As an immediate consequence, we prove the Maximal Symmetry Rank Conjecture for all $M\in \mathcal{M}_{0}^n$. Finally, we show the Maximal Symmetry Rank Conjecture for simply-connected, non-negatively curved manifolds holds for dimensions less than or equal to nine without assuming the torus action is almost isotropy-maximal or isotropy-maximal.

math.DG↗
arXiv2012

Topology of non-negatively curved manifolds

We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another are the circle bundles over a base, which itself is one of the family of CP^1 bundles over CP^2. We classify such manifolds up to diffeomorphism using the Kreck-Stolz invariants. Comparisons of the invariants is then used to find many diffeomorphism of the total space of these bundles with positively curved Eschenburg spaces. The total space of each bundle (in the case where the bundle is not spin) also admit a metric with non-negative sectional curvature, and in some cases an Einstein metric as well.

math.DG↗
arXiv2004

Topological properties of Eschenburg spaces and 3-Sasakian manifolds

The authors examine topological properties of the 7-dimensional Eschenburg biquotients diag(z^k1,z^k2,z^k3)\SU(3)/diag(z^l1,z^l2,z^l3). A subfamily of these spaces carry a 3-Sasakian metric. The authors show that among this subfamily there exist many 3-Sasakian spaces which are homeomorphic but not diffeomorphic. In addition, they construct a pair of 3-Sasakian spaces which are diffeomorphic, thus giving the first example of a manifold which carries two non-isometric 3-Sasakian metrics. This answers an open question of C. Boyer and K.Galicki. Among the general family, the authors construct many pairs of positively curved Eschenburg spaces which are homeomorphic but not diffeomorphic. Such pairs were first constructed by Kreck-Stolz among the special subfamily of Aloff-Wallach spaces.

math.DG↗