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Crichton Ogle

Publications and source records attributed to Crichton Ogle.

17 recordsLinked to original sources
arXiv2026

Pade Approximants for Geodesy

In this note we analyze the use of Padé approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.

math-ph↗
arXiv2022

On the chromatic localization of the homotopy completion tower for $\mathcal{O}$-algebras

The completion tower of a nonunital commutative ring is a classical construction in commutative algebra. In the setting of structured ring spectra as modeled by algebras over a spectral operad, the analogous construction is the homotopy completion tower. The purpose of this brief note is to show that localization with respect to the Johnson-Wilson spectrum $E(n)$ commutes with the terms of this tower.

math.AT↗
arXiv2021

On the structure of modules indexed by small categories

Given a small category C, a C-module M is a functor from C to the category of finite-dimensional vector spaces over a field k. Associated to M is its local structure, given as a functor from C to the category of bi-closed multi-flags over k. When the local structure of M is stable (a condition satisfied whenever both the category C and the field k are finite), it determines a quasi-tame cover QTC(M) (a finite direct sum of quasi-blocks), indexed by the same category, for which the associated graded local structure is canonically isomorphic to that of M. QTC(M) represents the closest approximation to M by a quasi-tame module, and recovers M precisely when M itself is quasi-tame. In the case M has stable local structure and is equipped with an inner product compatible with that structure, there exists a C-module surjection QTC(M) -> M inducing the above-mentioned isomorphism on associated graded local structures. This map is an isomorphism iff the excess of M vanishes (where the excess numerically measures the failure of the local structure of M to be in general position).

math.AT↗
arXiv2021

The Uniform Homotopy Category

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the respective Lipschitz and uniform settings. Cubical sets and uniform spaces admit the additional compatible structures of categories of (co)fibrant objects. A categorical equivalence between classical homotopy categories of cubical sets and spaces lifts to a full and faithful embedding from an associated Lipschitz homotopy category of cubical sets into an associated uniform homotopy category of uniform spaces. Bounded cubical cohomology generalizes to a representable theory on the Lipschitz homotopy category. Bounded singular cohomology on path-connected spaces generalizes to a representable theory on the uniform homotopy category. Along the way, this paper develops a cubical analogue of Kan's Ex^infinity functor and proves a cubical approximation theorem for uniform maps.

math.AT↗
arXiv2021

Invertibility in Category Representations

Inverse categories are categories in which every morphism x has a unique pseudo-inverse y in the sense that xyx=x and yxy=y. Persistence modules from topological data analysis and similarly decomposable category representations factor through inverse categories. This paper gives a numerical condition, decidable when the indexing category is finite, characterizing when a representation of a small category factors through an inverse category.

math.CT↗
arXiv2020

A Fundamental Theorem for the $K$-theory of connective $\mathbb{S}$-algebras

Invoking the density argument of Dundas-Goodwillie-McCarthy, we extend the Fundamental Theorem of $K$-theory from the category of simplicial rings to the category of $\mathbb{S}$-algebras. As an intermediate step, we prove the Fundamental Theorem for simplicial rings appealing to recent results from the first author's thesis. This recovers as a special case the Fundamental Theorem for the $K$-theory of spaces appearing in Hüttemann-Klein-Vogell-Waldhausen-Williams.

math.KT↗
arXiv2019

Relative Amenability and Relative Soficity

We define a notion of relative soficity for countable groups with respect to a family of groups. A group is sofic if and only if it is relative sofic with respect to the family consisting only of the trivial group. If a group is relatively sofic with respect to a family of sofic groups, then the group is sofic. Using this notion we generalize a theorem of Elek and Szabo on the soficity of an extension of sofic groups by amenable groups. In particular we prove that groups that are relatively sofic with respect to a family of sofic groups are sofic and more importantly extensions of sofic groups by residually amenable groups are sofic. We also construct examples of relatively amenable groups with respect to an infinite family of subgroups but not with respect to any finite subfamily of the subgroups. As an application of these constructions we show that Deligne's central extension groups are sofic.

math.GR↗
arXiv2017

$K$-theory of Hermitian Mackey functors and a reformulation of the Novikov Conjecture

We define a genuine $\mathbb{Z}/2$-equivariant real algebraic $K$-theory spectrum $KR(A)$, for every genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure. This construction extends the real $K$-theory of Hesselholt-Madsen for discrete rings and the Hermitian $K$-theory of Burghelea-Fiedorowicz for simplicial rings. We construct a natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of Bökstedt-Hsiang-Madsen. The trace provides a splitting of the real $K$-theory of the spherical group-ring. We use this splitting on the geometric fixed points of $KR$, which we regard as an $L$-theory of genuinely equivariant ring spectra, to reformulate the Novikov conjecture on the homotopy invariance of the higher signatures purely in terms of the module structure of the rational $L$-theory of the "Burnside group-ring".

math.AT↗
arXiv2017

A remark on the Isomorphism Conjectures

We show that for various natural classes of groups and appropriately defined K- and L-theoretic functors, injectivity or bijectivity of the assembly map follows from the Isomorphism Conjecture being true for acyclic groups lying within that class.

math.KT↗
arXiv2013

Strong embeddability and extensions of groups

We introduce the notion of strong embeddability for a metric space. This property lies between coarse embeddability and property A. A relative version of strong embeddability is developed in terms of a family of set maps on the metric space. When restricted to discrete groups, this yields relative coarse embeddability. We verify that groups acting on a metric space which is strongly embeddable has this relative strong embeddability, provided the stabilizer subgroups do. As a corollary, strong embeddability is preserved under group extensions.

math.MG↗
arXiv2013

Totalization of simplicial homotopy types

We identify the obstructions for the functoriality and the uniqueness of the totalization functor, (partially) defined on the category of simplicial objects in the homotopy category of a stable model category, and we use a result from the cyclic homology of group algebras to show they can be non-zero.

math.AT↗
arXiv2012

Asymptotically exact spaces and coarse assembly

Between the category of exact metric spaces with bounded geometry (about which much is known) and the larger category of arbitrary exact metric spaces (about which little is known) lies the intermediate category of asymptotically exact metric spaces. We show that the coarse Baum-Connes assembly map is naturally split surjective for this class, with generally non-zero kernel.

math.GT↗
arXiv2012

On the Hochschild and cyclic (co)homology of rapid decay group algebras

We show that the technical condition of solvable conjugacy bound, introduced in \cite{JOR1}, can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands $HH_*^t(\BG)_{ }$ and $HC_*^t(\BG)_{ }$ for any bounding class $\B$, discrete group with word-length $(G,L)$ and conjugacy class $ \in $. We use this description to prove the conjecture $\B$-SrBC of \cite{JOR1} for a class of groups that goes well beyond the cases considered in that paper. In particular, we show that the conjecture $\ell^1$-SrBC (the Strong Bass Conjecture for the topological $K$-theory of $\ell^1(G)$) is true for all semihyperbolic groups which satisfy SrBC, a statement consistent with the rationalized Bost conjecture for such groups.

math.KT↗
arXiv2011

Relative property A and relative amenability for countable groups

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if a group has property A relative to a family of subgroups, each of which has property A, then the group has property A. This result leads to new classes of groups that have property A. In particular, groups are of property A if they act cocompactly on locally finite property A spaces of bounded geometry with at least one stabilizer of property A. Specializing the definition of relative property A, an analogue definition of relative amenability for discrete groups are introduced and similar results are obtained.

math.GR↗
arXiv2011

Filtrations of simplicial functors and the Novikov Conjecture

We show that the Strong Novikov Conjecture for the maximal C*-algebra C*(G) of a discrete group G is equivalent to a statement in topological K-theory for which the corresponding statement in algebraic K-theory is always true. We also show that for any group G, rational injectivity of the full assembly map for the topological K-theory of C*(G) follows from rational injectivity of the restricted assembly map.

math.KT↗