arXiv subjects

Rekha Santhanam

Publications and source records attributed to Rekha Santhanam.

16 recordsLinked to original sources

arXiv2026

Brown's Asymptotic Limit Functor and Proper Homology

In 1974, E. M. Brown introduced the $\wp$-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that $\wp$ has major advantages over the classical limits $\varprojlim$ and $\varprojlim^1$: it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown--Grossman proper fundamental group of any open contractible $3$-manifold other than $\mathbb{R}^3$ is uncountable and perfect.

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arXiv2026

Equivariant Relative Sectional Category and Induced Invariants

The relative sectional category, introduced by González, Grant, and Vandembroucq for fibrations and later extended by García-Calcines to arbitrary maps, provides a common framework encompassing several numerical homotopy invariants, including the Lusternik--Schnirelmann category, the topological complexity of a map, and homotopic distance. In this paper, we introduce and study the equivariant analogue of the relative sectional category for $G$-maps. We establish its fundamental homotopy-theoretic properties, including comparison, product, and composition inequalities, as well as its behavior under changes of domain and codomain. As applications, we introduce and investigate equivariant analogues of the topological complexity of a map, in the sense of Scott and Murillo--Wu, and the equivariant Lusternik--Schnirelmann category of a map. Several examples are provided to illustrate the theory and demonstrate that these invariants extend the corresponding classical equivariant notions.

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arXiv2026

Higher topological complexity of Seifert fibered manifolds

In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r^{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$, and characterize large families where the value is $3r$ or $3r+1$. Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is exactly $3r+1$.

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arXiv2025

Characterizing model structures on finite posets

Transfer systems on finite posets have recently been gaining traction as a key ingredient in equivariant homotopy theory. Additionally, they also naturally occur in the data of a model structure. We give a complete characterization of all model category structures on a finite lattice, using transfer systems as our main tool, resulting in new connections between abstract homotopy theory and equivariant methods.

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arXiv2025

Equivariant Intrinsic Formality

Algebraic models for equivariant rational homotopy theory were developed by Triantafillou and Scull for finite group actions and $S^1$ action, respectively. They showed that given a diagram of rational cohomology algebras from the orbit category of a group $G$, there is a unique minimal system of DGAs and hence a unique equivariant rational homotopy type that is weakly equivalent to it. However, there can be several equivariant rational homotopy types with the same system of cohomology algebras. Halperin, Stasheff, and others studied the problem of classifying rational homotopy types up to cohomology in the non-equivariant case. In this article, we consider this question in the equivariant case. We prove that when $\mathbb{Z}_p$ under suitable conditions, the equivariant rational homotopy types with isomorphic cohomology can be reduced to the non-equivariant case.

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arXiv2024

Cofibrantly generated model structures for functor calculus

Model structures for many different kinds of functor calculus can be obtained by applying a theorem of Bousfield to a suitable category of functors. In this paper, we give a general criterion for when model categories obtained via this approach are cofibrantly generated. Our examples recover the homotopy functor and $n$-excisive model structures of Biedermann and Röndigs, with different proofs, but also include a model structure for the discrete functor calculus of Bauer, Johnson, and McCarthy.

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arXiv2024

Uniquely compatible transfer systems for cyclic groups of order $p^rq^s$

Bi-incomplete Tambara functors over a group $G$ can be understood in terms of compatible pairs of $G$-transfer systems. In the case of $G = C_{p^n}$ , Hill, Meng and Li gave a necessary and sufficient condition for compatibility and computed the exact number of compatible pairs. In this article, we study compatible pairs of $G$-transfer systems for the case $G = C_{p^rq^s}$ and identify conditions when such transfer systems are uniquely compatible in the sense that they only form trivially compatible pairs. This gives us new insight into collections of norm maps that are relevant in equivariant homotopy theory.

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arXiv2024

Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs

The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number also plays an important role in a variety of Helly type results. We show that the non-cover complex of a hypergraph $\mathcal{H}$ is $|V(\mathcal{H)}|- γ_i(\mathcal{H})-1$-collapsible, where $γ_i(\mathcal{H})$ is the generalization of independence domination number of a graph to hypergraph. This extends the result of Choi, Kim and Park from graphs to hypergraphs. Moreover, the upper bound in terms of strong independence domination number given by Kim and Kim for the Leray number of the non-cover complex of a hypergraph can be obtained as a special case of our result. In general, there can be a large gap between the collapsibility number of a complex and its well-known upper bounds. In this article, we construct a sequence of upper bounds $\mathcal{M}_k(X)$ for the collapsibility number of a simplicial complex $X$, which lie in this gap. We also show that the bound given by $\mathcal{M}_k$ is tight if the underlying complex is $k$-vertex decomposable.

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arXiv2023

Cofibration category structures on the category of graphs

In this article, we show that there is no cofibration category structure on the category of finite graphs with $\times$-homotopy equivalences as the class of weak equivalences. Further, we show that it is not possible to enlarge the class of weak equivalences to get cofibration category structure on the category of finite graphs without including morphisms where domain and codomain have non-isomorphic stiff subgraphs.

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arXiv2023

Vertex cut of a graph and connectivity of its neighbourhood complex

We show that if a graph $G$ satisfies certain conditions then the connectivity of neighbourhood complex $\mathcal{N}(G)$ is strictly less than the vertex connectivity of $G$. As an application, we give a relation between the connectivity of the neighbourhood complex and the vertex connectivity for stiff chordal graphs, and for weakly triangulated graphs satisfying certain properties. Further, we prove that for a graph $G$ if there exists a vertex $v$ satisfying the property that for any $k$-subset $S$ of neighbours of $v$, there exists a vertex $v_S \neq v$ such that $S$ is subset of neighbours of $v_S$, then $\mathcal{N}(G-\{v\})$ is $(k-1)$-connected implies that $\mathcal{N}(G)$ is $(k-1)$-connected. As a consequence of this, we show that:(i) neighbourhood complexes of queen and king graphs are simply connected and (ii) if $G$ is a $(n+1)$-connected chordal graph which is not folded onto a clique of size $n+2$, then $\mathcal{N}(G)$ is $n$-connected.

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arXiv2022

Enriched functor categories for functor calculus

In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.

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arXiv2021

(Lack of) Model Structures on the Category of Graphs

In this article, we study model structures on the category of finite graphs with $\times$-homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Strøm-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with $\times$-homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions.

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arXiv2019

Lovász' original lower bound: Getting tighter bounds and Reducing computational complexity

In this article, we give conditions on a graph under which the Lovász' original bound of the graph can be improved by increasing the topological connectivity of its neighbourhood complex. We also work out conditions under which computing the topological connectivity of hom complex of a pair of graphs can be simplified. In particular, hom complex as a covariant functor acting on a double mapping cylinder of graphs is a homotopy pushout of hom complex functor applied to its subgraphs. We give applications of this result where the computation of hom complexes is simplified. Finally, we explain why double mapping cylinder of graphs does not give a satisfactory definition of homotopy pushout in the category of graphs.

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arXiv2015

A short treatise on Equivariant Gamma spaces

Equivariant $Γ$-spaces model equivariant infinite loop spaces. In this article, we show that there exists a connective Quillen equivalence between the category of equivariant $Γ$-spaces and the category of orthogonal spectra.

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arXiv2012

Sheaves and $K$-theory for $\mathbb{F}_1$-schemes

This paper is devoted to the open problem in $\mathbb{F}_1$-geometry of developing $K$-theory for $\mathbb{F}_1$-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for $\mathcal{M}_0$-schemes and $\mathbb{F}_1$-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over $\mathbb{F}_1$. Special attention is paid to two aspects particular to $\mathbb{F}_1$-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillen's Q-construction to a definition of $G$-theory and $K$-theory for $\mathbb{F}_1$-schemes. A comparison with Waldhausen's $S_{\bullet}$-construction yields the ring structure of $K$-theory. In particular, we generalize Deitmar's $K$-theory of monoids and show that $K_*(\Spec\mathbb{F}_1)$ realizes the stable homotopy of the spheres as a ring spectrum.

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arXiv2011

Units of equivariant ring spectra

It is well known that very special $Γ$-spaces and grouplike $\E_\infty$ spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $Γ$-spaces and showed that special equivariant $Γ$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant $\E_\infty$-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $Γ$-spaces is Quillen equivalent to the category of equivariant $\E_\infty$ spaces. We define the units of equivariant ring spectra in terms of equivariant $Γ$-spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

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