arXiv subjects

Carmen Rovi

Publications and source records attributed to Carmen Rovi.

14 recordsLinked to original sources

arXiv2026

A normal form for the Thin Flat Surfaces category

The category TFS of thin flat surfaces, introduced by Khovanov, Qi, and Rozansky, is a strict symmetric monoidal category whose morphisms are compact oriented surfaces with corners arising as neighbourhoods of immersed graphs in a strip. We prove two main results. First, a normal form theorem: every connected viewable tf-cobordism is equal, in TFS, to a canonical composite determined by its topological type. Second, a sufficiency theorem: we establish a list of relations in the TFS category which are sufficient, meaning that any two composites of generators representing the same tf-cobordism are related by a finite sequence of the listed relations. Together, these results give a complete generators-and-relations presentation of TFS.

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arXiv2026

A Stable Distance Persistence Homology for Dynamic Bayesian Network Clustering

Dynamic Bayesian networks (DBNs) are a widely used framework for modeling systems whose probabilistic structure evolves over time. Standard inference methods focus on local conditional distributions and can miss larger-scale patterns in how dependencies between variables organize and change over time. We introduce a topological approach to this problem. To each DBN we associate a time-varying graph, called a Dynamic Bayesian Graph (DBG), by assigning to each edge a strength that measures variation in its conditional dependence across parent configurations, and retaining edges whose strength exceeds a chosen threshold. We show that this construction fits within the dynamic graph framework of Kim and Mémoli, enabling the use of tools from topological data analysis. Applying persistent homology to a DBG produces a barcode, which records the merging and disappearance of connected groups of strongly dependent variables over time. We prove that this barcode is stable: small perturbations in the conditional probability tables of the DBN lead to small changes in the resulting barcode. This yields a principled and noise-resistant summary of how dependency structure evolves in a dynamic Bayesian network.

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arXiv2025

Algebraic Structures in Microtonal Music

We will discuss how certain group theory structures are found in music theory. Western music splits the octave into 12 equal tones called half-steps. We can take this division further and split the octave into 24 equal tones by splitting each half-step in two, called a quarter-step. By assigning each of these 24 notes a number, we can discuss musical actions mathematically. In this paper, we analyze 24-tone microtonal music and explore how musical and harmonic structures in this system can be interpreted in terms of group-theoretic structures. This work extends the study by Crans, Fiore, and Satyendra.

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arXiv2025

Nested cobordisms, Cyl-objects and Temperley-Lieb algebras

We introduce a discrete cobordism category for nested manifolds and nested cobordisms between them. A variation of stratified Morse theory applies in this case, and yields generators for a general nested cobordism category. Restricting to a low-dimensional example of the ``striped cylinder'' cobordism category Cyl, we give a complete set of relations for the generators. With an eye towards the study of TQFTs defined on a nested cobordism category, we describe functors Cyl$\to\mathcal{C}$, which we call Cyl-objects in $\mathcal{C}$, and show that they are related to known algebraic structures such as Temperley-Lieb algebras and cyclic objects. We moreover define novel algebraic constructions inspired by the structure of Cyl-objects, namely a doubling construction on cyclic objects analogous to edgewise subdivision, and a cylindrical bar construction on self-dual objects in a monoidal category.

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arXiv2024

A K-theory spectrum for cobordism cut and paste groups

Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group $\overline{\mathrm{SK}}_n$ is defined. In this paper, we extend this definition to manifolds with boundary obtaining $\overline{\mathrm{SK}}^{\partial}_n$ and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on $π_0$ the cobordism cut and paste groups of manifolds with boundary, $\overline{\mathrm{SK}}^{\partial}_n$, and a map of spectra that lifts the canonical quotient map $\mathrm{SK}^{\partial}_n \rightarrow \overline{\mathrm{SK}}^{\partial}_n$.

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arXiv2023

Chain duality for categories over complexes

We show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference, is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincaré duality to global Poincaré duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.

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arXiv2021

Cut and paste invariants of manifolds via algebraic K-theory

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic $K$-theory, and applying these tools to studying the Grothendieck ring of varieties. In this paper we give a new application of their framework: we construct a $K$-space that recovers the classical $\mathrm{SK}$ ("schneiden und kleben," German for "cut and paste") groups for manifolds on $π_0$, and we construct a derived version of the Euler characteristic.

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arXiv2020

Signature Cocycles on the Mapping Class Group and Symplectic Groups

Werner Meyer constructed a cocycle in $H^2(Sp(2g, \mathbb{Z}); \mathbb{Z})$ which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study the signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer N. Using these results, we are able to give a complete answer for N = 2, 4 and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.

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arXiv2018

Cohomology of symplectic groups and Meyer's signature theorem

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z} \to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to 1$ the pullback of the universal cover of $\mathsf{ Sp}(2g,\mathbb{R})$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g, \mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf {Sp}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}/2\to\tilde{\mathfrak{H}}\to\mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$. It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\widetilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.

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arXiv2018

Relating Cut and Paste Invariants and TQFTs

In this paper, we shall be concerned with a relation between TQFTs and cut and paste invariants introduced by Karras, Kreck, Neumann and Ossa. Cut and paste invariants, or SK invariants, are functions on the set of smooth manifolds that are invariant under the cutting and pasting operation. Central to the work in this paper are also SKK invariants, whose values on cut and paste equivalent manifolds differ by an error term depending only on the glueing diffeomorphism. Here we investigate a surprisingly natural group homomorphism between the group of invertible TQFTs and the group of SKK invariants and describe how these groups fit into an exact sequence. We conclude in particular that all positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs. All manifolds are smooth and oriented throughout unless stated otherwise.

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arXiv2017

Hirzebruch $χ_y$-genera modulo $8$ of fiber bundles for odd integers $y$

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total space are compact complex algebraic manifolds (in the complex analytic topology, not in the Zariski topology). We show that the Hirzebruch $χ_y$-genera of such a fibre bundle are always multiplicative modulo 4. We also investigate multiplicativity modulo 8 and show that if $y$ is congruent to 3 modulo 4, then the $χ_y$-genera are multiplicative modulo 8. We also show that when $y$ is congruent to 1 modulo 4, the Hirzebruch $χ_y$-genera of such a fiber bundle are multiplicative modulo 8 if and only if the signature is multiplicative modulo 8, and that the non-multiplicativity modulo 8, in this case, is identified with an Arf-Kervaire invariant.

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arXiv2017

The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4. In this paper we present two results concerning the multiplicativity modulo 8: firstly we identify the obstruction to multiplicativity modulo 8 with the Arf-Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if the fibre is even-dimensional and the action of the fundamental group of the base is trivial on the middle cohomology of the fibre with $\mathbb{Z}_4$ coefficients, then this Arf-Kervaire invariant takes value 0 and hence the signature is multiplicative modulo 8.

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arXiv2015

The signature modulo 8 of fibre bundles

This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a Z_2-valued quadratic form. This result is applied to a fibration of geometric Poincare complexes with the total space 4k-dimensional. It is known that the signature is multiplicative modulo 4, and that there are examples of non-multiplicativity modulo 8. We identify the obstruction to multiplicativity modulo 8 with the Arf invariant of the Z_2-valued quadratic form defined on an appropriate Z_2-cohomology vector space by dividing the Z_4-valued Pontryagin square by 2. The obstruction is then shown to vanish under certain assumptions.

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arXiv2014

Orbispaces and their Mapping Spaces via Groupoids: A Categorical Approach

In this paper, we give an accessible introduction to the theory of orbispaces via groupoids. We define a certain class of topological groupoids, which we call orbigroupoids. Each orbigroupoid represents an orbispace, but just as with orbifolds and Lie groupoids, this representation is not unique: orbispaces are Morita equivalence classes of orbigroupoids. We show how to formalize this equivalence by defining the category of orbispaces as a bicatecory of fractions from the category of orbigroupoids. We focus particularly on laying the groundwork for future work in creating mapping objects for orbispaces which are themselves orbispaces, and providing a concrete description of how this mapping space construction will get its orbispace structure. Throughout this paper, we illustrate our definitions and results with numerous examples which we hope will be useful in seeing how the categorical point of view is used to study these spaces.

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