Searcharxiv⌕ Search

arXiv subjects

Donald Stanley

Publications and source records attributed to Donald Stanley.

At least 19 recordsLinked to original sources
arXiv2026

Moment angle complexes and duality for tight manifolds

For a field $\mathbb{F}$ and a triangulated compact $\mathbb{F}$-orientable manifold, consider the homology of the associated Moment-Angle ccomplex $H_*(\mathcal{Z}_{\mathcal{K}})$. We show the total homology rank $\beta(\mathcal{Z}_{\mathcal{K}})$ satisfies the inequality $\beta(\mathcal{Z}_{\mathcal{K}};\mathbb{F})\geq 2^{m-1}(\beta(\mathcal{K};\mathbb{F})-2)+2$, with equality occurring exactly when the triangulation is $\mathbb{F}$-tight. Using Lefschetz duality, we introduce a short exact sequence of functors that, in turn, introduces a new duality theorem in Double Homology for tight manifold triangulations.

math.AT↗
arXiv2025

Realizing orders in rational sphere product algebras with three generators

The realization problem asks which algebras can be realized as the cohomology of spaces. We study this problem in the context of the orders in a graded rational exterior algebra on three generators. An order is a subring whose underlying additive group is a lattice. We give conditions for when such an order is realizable, and in particular show that in the simply-connected case any order is realizable if the generators of the exterior algebra are of odd degree.

math.RA↗
arXiv2025

Realization of some Stanley-Reisner algebras and graph colorings

It is a classical problem in algebraic topology to decide whether a given graded $\mathbb{Z}$-algebra can be realized as the cohomology ring of a space. In this paper, we introduce families of Stanley-Reisner algebras depending on graphs, and relate their realizability to the span coloring of the graph.

math.AT↗
arXiv2025

Graph colouring and Steenrod's problem for Stanley-Reisner rings

It is a classical problem in algebraic topology asked by Steenrod which graded rings occur as the cohomology ring of a space. In this paper, we define an algebraic version of the graph colouring, span colouring, and observe the relation between span colourings and Steenrod's problem for graded Stanley-Reisner rings, in other words polynomial rings divided by an ideal generated by square-free monic monomials.

math.AT↗
arXiv2023

Double cohomology of moment-angle complexes

We put a cochain complex structure ${CH}^*(\mathcal Z_K)$ on the cohomology of a moment-angle complex $\mathcal Z_K$ and call the resulting cohomology the double cohomology, ${HH}^*(\mathcal Z_K)$. We give three equivalent definitions for the differential, and compute ${HH}^*(\mathcal Z_K)$ for a family of simplicial complexes containing clique complexes of chordal graphs.

math.AT↗
arXiv2023

Weighted polyhedral products and Steenrod's problem

We construct a weighted version of polyhedral products and compute its cohomology in special cases. This is applied to resolve Steenrod's cohomology realization problem in a case related to products of spheres.

math.AT↗
arXiv2023

A stability theorem for bigraded persistence barcodes

We define bigraded persistent homology modules and bigraded barcodes of a finite pseudo-metric space X using the ordinary and double homology of the moment-angle complex associated with the Vietoris-Rips filtration of X. We prove a stability theorem for the bigraded persistent double homology modules and barcodes.

math.AT↗
arXiv2022

Multiparameter persistence modules in the large scale

A persistence module with $m$ discrete parameters is a diagram of vector spaces indexed by the poset $\mathbb{N}^m$. If we are only interested in the large scale behavior of such a diagram, then we can consider two diagrams equivalent if they agree outside of a ``negligeable'' region. In the $2$-dimensional case, we classify the indecomposable diagrams up to finitely supported diagrams. In higher dimension, we partially classify the indecomposable diagrams up to suitably finite diagrams, and show that the full classification problem is wild.

math.AT↗
arXiv2022

The Fundamental Group of Torus Knots

This work is concerned with the calculation of the fundamental group of torus knots. Torus knots are special types of knots which wind around a torus a number of times in the longitudinal and meridional directions. We compute and describe the fundamental group of torus knots by using some concepts in algebraic topology and group theory. We also calculate the fundamental group of an arbitrary knot by using an algorithm called the Wirtinger presentation.

math.GT↗
arXiv2021

Realization of graded monomial ideal rings modulo torsion

Let $A$ be the quotient of a graded polynomial ring $\mathbb{Z}[x_1,\cdots,x_m]\otimesΛ[y_1,\cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.

math.AT↗
arXiv2020

Generating functions and topological complexity

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the pole equals $\cat(X)$. Here $X$ is a finite CW-complex and for $r\ge 2$ the symbol $\tc_r(X)$ denotes its $r$-th sequential topological complexity. We analyse an example (violating the Ganea conjecture) and conclude that part (b) of the rationality conjecture is false in general. Besides, we establish a cohomological version of the rationality conjecture.

math.AT↗
arXiv2018

Cohomology Rings of a Class of Torus Manifolds

Torus manifolds are topological generalization of smooth projective toric manifolds. We compute the rational cohomology ring of a class of smooth locally standard torus manifolds whose orbit space is a connected sum of simple polytopes.

math.AT↗
arXiv2018

Exact weights, path metrics, and algebraic Wasserstein distances

We use weights on objects in an abelian category to define what we call a path metric. We introduce three special classes of weight: those compatible with short exact sequences; those induced by their path metric; and those which bound their path metric. We prove that these conditions are in fact equivalent, and call such weights exact. As a special case of a path metric, we obtain a distance for generalized persistence modules whose indexing category is a measure space. We use this distance to define Wasserstein distances, which coincide with the previously defined Wasserstein distances for one-parameter persistence modules. For one-parameter persistence modules, we also describe maps to and from an interval module, and we give a matrix reduction for monomorphisms and epimorphisms.

math.RA↗
arXiv2018

Polynomial functors in manifold calculus

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F: O(M)--> Top from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree <= k are determined by their values on O_k(M), where O_k(M) is the full subposet of O(M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace O_k(M) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O(M) to any simplicial model category C. If conf(k, M) stands for the unordered configuration space of k points in M, we also show that the category of homogeneous cofunctors O(M) --> C of degree k is weakly equivalent to the category of linear cofunctors O(conf(k, M)) --> C provided that C has a zero object. Using a completely different approach, we also show that if C is a general model category and F: O_k(M) --> C is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion O_k(M) --> O(M) is also an isotopy cofunctor.

math.AT↗
arXiv2018

Classification of homogeneous functors in manifold calculus

For any object A in a simplicial model category M, we construct a topological space \^A which classifies homogeneous functors whose value on k open balls is equivalent to A. This extends a classification result of Weiss for homogeneous functors into topological spaces.

math.AT↗
arXiv2018

Very good homogeneous functors in manifold calculus

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Let C be a category that has a zero object and all small limits. A homogeneous functor (in the sense of manifold calculus) of degree k from O(M) to C is called very good if it sends isotopy equivalences to isomorphisms. In this paper we show that the category VGHF of such functors is equivalent to the category of contravariant functors from the fundamental groupoid of Conf(k, M) to C, where Conf(k, M) stands for the unordered configuration space of k points in M. As a consequence of this result, we show that the category VGHF is equivalent to the category of representations of the fundamental group of Conf(k, M) in C, provided that Conf(k, M) is connected. We also introduce a subcategory of vector bundles that we call very good vector bundles, and we show that it is abelian, and equivalent to a certain category of very good functors.

math.AT↗