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Jelena Grbić

Publications and source records attributed to Jelena Grbić.

15 recordsLinked to original sources
arXiv2024

Steenrod operations on polyhedral products

We describe the action of the mod $2$ Steenrod algebra on the cohomology of various polyhedral products and related spaces. We carry this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, we deduce some consequences for the lowest cohomological dimension in which non-trivial Steenrod operations can appear. We present a version of cochain-level formulas for Steenrod operations on simplicial complexes. We explain the idea of "propagating" such formulas from a simplicial complex $K$ to polyhedral joins over $K$ and we give examples of this process. We tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes. Although these are cases where one can understand the Steenrod action via a stable homotopy decomposition, we anticipate applying this method to cases where there is no such decomposition.

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arXiv2022

The cohomology of the free loop spaces of $SU(n+1)/T^n$

We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the cohomology algebra of the free loop spaces. We build new theory to allow for the computation of $H^*(Λ(SU(n+1)/T^{n});\mathbb{Z})$, a significantly more complicated structure than other known examples. In addition to our theoretical results, we explicitly implement a novel integral Gröbner basis procedure for computation. This procedure is applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. The power of this procedure is illustrated by the explicit calculation of $H^*( Λ(SU(4)/T^3);\mathbb{Z})$. We also provide a python library with examples of all procedures used in the paper.

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arXiv2022

Models for the Cohomology of Certain Polyhedral Products

For a commutative ring $\mathbf k$ with unit, we describe and study various differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are models for the cohomology of polyhedral products $(\underline{CX},\underline X)^K$. Along the way, we prove that the integral cohomology $H^*((D^1, S^0)^K; \mathbb Z)$ of the real moment-angle complex is a Tor module, the one that does not come from a geometric setting. We also reveal that the apriori different cup product structures in $H^*((D^1, S^0)^K;\mathbb Z)$ and in $H^*((D^n, S^{n-1})^K; \mathbb Z)$ for $n\geq 2$ have the same origin. As an application, this work sets the stage for studying the based loop space of $(\underline{CX}, \underline X)^K$ in terms of the bar construction applied to the differential graded $\mathbb Z$-algebras $B(\mathcal C^*(\underline X; \mathbb Z), K) $ quasi-isomorphic to the singular cochain algebra $\mathcal C^*((\underline{CX},\underline X)^K;\mathbb Z)$.

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arXiv2022

One-relator groups and algebras related to polyhedral products

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal{R}_K$, to be a one-relator group; and for the Pontryagin algebra $H_*(Ω\mathcal{Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterisation of these properties. For $RC_K'$, it is given by a condition on the homology group $H_2(\mathcal{R}_K)$, whereas for $H_*(Ω\mathcal{Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal{Z}_K$.

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arXiv2022

Duality in Toric Topology

We characterise integral Poincaré duality moment-angle complexes $\mathcal{Z}_{\mathcal{K}}$ in combinatorial terms of the Fan-Wang duality of the simplicial complex $\mathcal{K}$, and consequently in algebraic terms of the Gorenstein duality of the Stanley-Reisner ring $\mathbb{Z}[\mathcal{K}]$. We extend Poincaré duality results to certain polyhedral products using polyhedral join products of simplicial complexes.

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arXiv2021

Normalizers of chains of discrete $p$-toral subgroups in compact Lie groups

In this paper we study the normalizer decomposition of a compact Lie group $G$ using the information of the fusion system $\mathcal{F}$ of $G$ on a maximal discrete $p$-toral subgroup. We prove that there is an injective map from the set of conjugacy classes of chains of $\mathcal{F}$-centric, $\mathcal{F}$-radical discrete $p$-toral subgroups to the set of conjugacy classes of chains of $p$-centric, $p$-stubborn continuous $p$-toral subgroups. The map is a bijection when $π_0(G)$ is a finite $p$-group. We also prove that the classifying space of the normalizer of a chain of discrete $p$-toral subgroups of $G$ is mod $p$ equivalent to the classifying space of the normalizer of the corresponding chain of $p$-toral subgroups.

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arXiv2021

Non-trivial higher Massey products in moment-angle complexes

As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of infinitely many non-formal manifolds that appear in topology, complex geometry and algebraic geometry.

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arXiv2021

A Unified Topological Approach to Data Science

We establish a new theory which gives a unified topological approach to data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

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arXiv2021

$LS$-category of moment-angle manifolds and higher order Massey products

Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the combinatorics of $K$, the cup product length of $H^*(\zk)$, as well as a certain Massey products. We show that the LS category closely relates to vanishing of Massey products in $H^*(\zk)$ and through this connection we describe first structural properties of Massey products in moment-angel manifolds. Some of further applications include calculations of the LS category and the description of conditions for vanishing of Massey products for moment-angle manifolds over fullerenes, Pogorelov polytopes and $k$-neighbourly complexes, which double as important examples of hyperbolic manifolds.

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arXiv2020

A new approach to mod 2 decompositions of BSU(2) and BSO(3)

Dwyer, Miller and Wilkerson proved that at the prime 2, the classifying spaces of SU(2) and SO(3) can be obtained as a homotopy pushout of the classifying spaces of certain subgroups. In this paper we show explicitly how these decompositions arise from the fusion systems of SU(2) and SO(3) over maximal discrete 2-toral subgroups.

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arXiv2019

Lowest-degree triple Massey products in moment-angle complexes

We give a combinatorial classification of non-trivial triple Massey products of three dimensional classes in the cohomology of a moment-angle complex. This work improves on a result by Denham and Suciu (2007) by considering triple Massey products with non-trivial indeterminacy.

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arXiv2019

Simplicial $G$-complexes and representation stability of polyhedral products

Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [2] of $Σ(X, A)^K$ is then $G$-equivariant after suspension. In the case of $Σ_m$-polyhedral products, we give criteria on simplicial $Σ_m$-complexes which imply representation stability of $Σ_m$-representations $\{H_i((X, A)^{K_m})\}$.

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arXiv2018

The cohomology of free loop spaces of rank 2 flag manifolds

A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are $SU(3)/T^2$, $Sp(2)/T^2$, $Spin(4)/T^2$, $Spin(5)/T^2$ and $G_2/T^2$. In this paper we calculate the cohomology of the free loop space of rank 2 complete flag manifolds.

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arXiv2017

Configuration Spaces and Polyhedral Products

This paper aims to find the most general combinatorial conditions under which a moment-angle complex $(D^2,S^1)^K$ is a co-$H$-space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment-angle complex over a Golod simplicial complex is a co-$H$-space. Our main tool is a certain generalisation of the theory of labelled configuration spaces.

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