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Vicente Muñoz

Publications and source records attributed to Vicente Muñoz.

At least 19 recordsLinked to original sources
arXiv2026

Realizing additive monoids as mapping degree sets

We prove that mapping degree sets are stable under multiplication by finite subsets of $\mathbb Z$ containing $0$ and by sets obtained from additive submonoids of $\mathbb Z$ through finitely many sums and products. In particular, every set of the latter type occurs as a mapping degree set. As a consequence, we obtain a broad family of infinite mapping degree sets, including finite unions of arithmetic progressions starting at $0$. These results extend previous work on the realization problem and are related to a question posed by Neofytidis, Wang, and Wang.

math.AT↗
arXiv2026

Stirling-Ramanujan constants are exponential periods

Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral representations for all these constants that appear as exponential periods in the field $\mathbb Q (t,e^{-t})$ which reveals their natural transalgebraic nature. We conjecture that all these constants are transcendental numbers. Euler-Mascheroni's and Stirling's integral formula are classical, but the integral formula for Glaisher-Kinkelin appears to be new, as well as the integral formulas for the higher Stirling-Ramanujan constants. The method presented generalizes naturally to prove that many other constants are exponential periods over the field $\mathbb Q(t,e^{-t})$.

math.NT↗
arXiv2026

On the construction of K-contact non-Sasakian Smale-Barden manifolds

In the breakthrough paper \cite{Mu-jems}, it is constructed the first example of a simply connected compact $5$-manifold (aka.\ Smale-Barden manifold) which admits a K-contact structure but does not carry a Sasakian structure. In this work we clarify some aspects of the construction of \cite{Mu-jems}, determining explicitly the number $N$ of symplectic surfaces needed to have an isotropy locus that produce a $5$-manifold that is K-contact but not Sasakian. Also, in order to analyse the geography problem of determining which Smale-Barden manifolds admit K-contact but not Sasakian structures, we refine and generalize the constructions of symplectic surfaces in a symplectic $4$-manifold with transversal intersections giving rise to such manifolds.

math.DG↗
arXiv2025

Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants

This paper reinterprets Alexander-type invariants of knots via representation varieties of knot groups into the group $\textrm{AGL}_1(\mathbb{C})$ of affine transformations of the complex line. In particular, we prove that the coordinate ring of the $\textrm{AGL}_{1}(\mathbb{C})$-representation variety is isomorphic to the symmetric algebra of the Alexander module. This yields a natural interpretation of the Alexander polynomial as the singular locus of a coherent sheaf over $\mathbb{C}^*$, whose fibres correspond to quandle representation varieties of the knot quandle. As a by-product, we construct Topological Quantum Field Theories that provide effective computational methods and recover the Burau representations of braids. This theory offers a new geometric perspective on classical Alexander invariants and their functorial quantization.

math.GT↗
arXiv2025

The local moduli space of the Einstein-Yang-Mills system

We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth slice in the tame Fréchet category for the coupled action of bundle automorphisms on metrics and connections. Using this result, together with a careful analysis of the linearization of the Einstein-Yang-Mills system, we realize the moduli space of Einstein-Yang-Mills pairs modulo automorphism as an analytic set in a finite-dimensional tame Fréchet manifold, extending classical results of Koiso for Einstein metrics and Yang-Mills connections to the Einstein-Yang-Mills system. Furthermore, we introduce the notion of \emph{essential deformation} of an Einstein-Yang-Mills pair, which we characterize in full generality and explore in more detail in the four-dimensional case, proving a decoupling result for trace deformations when the underlying Einstein-Yang-Mills pair is a Ricci-flat metric coupled to an anti-self-dual instanton. In particular, we find a novel obstruction that does not occur in the \emph{decoupled} Einstein or Yang-Mills moduli problems. Finally, we prove that every essential deformation of the four-dimensional Einstein-Yang-Mills system based on a Calabi-Yau metric coupled to an instanton is of restricted type.

math.DG↗
arXiv2025

Constructions of symplectic surfaces in symplectic 4-manifolds with transversal intersections

In the breakthrough paper [V. Muñoz, A Smale-Barden manifold admitting K-contact but not Sasakian structure, 2024, 10.4171/JEMS/1496], it is constructed the first example of a simply connected compact 5-manifold (aka.\ Smale-Barden manifold) which admits a K-contact structure but does not carry a Sasakian structure, thus settling the question raised as Open Problem 10.2.1 in n [C. Boyer and K. Galicki, Sasakian Geometry, OUP, 2007]. In this paper we revise, refine and generalize the constructions of symplectic surfaces in a symplectic 4-manifold with transversal intersections. These are needed to produce the ramification locus of Seifert bundles over symplectic 4-orbifolds that serve to produce K-contact 5-manifolds.

math.SG↗
arXiv2024

Finite sets containing zero are mapping degree sets

In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the rational setting, where an affirmative answer is also given.

math.GT↗
arXiv2024

Character varieties of torus links

In this paper, we study the geometry of the moduli space of representations of the fundamental group of the complement of a torus link into an algebraic group G, an algebraic variety known as the G-character variety of the torus link. These torus links are a family of links in the 3-dimensional sphere formed by stacking several copies of torus knots. We develop an intrinsic stratification of the variety that allows us to relate its geometry with the one of the underlying torus knot. Using this information, we explicitly compute the E-polynomial associated to the Hodge structure of these varieties for $G=SL_2(\mathbb{C})$ and $SL_3(\mathbb{C})$, for an arbitrary torus link, showing an unexpected relation with the number of strands of the link.

math.GT↗
arXiv2024

Representation varieties of twisted Hopf links

We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties.

math.GT↗
arXiv2023

Nearly parallel $G_2$-manifolds: formality and associative submanifolds

We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over $CP^2$ with fibre $S^1\times S^2$ or $S^3/Z_p$ ($p>0$), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal parallel $Spin(7)$-manifolds. On the other hand, we construct associative minimal submanifolds in the Aloff-Wallach spaces and in any regular Sasaki-Einstein 7-manifold; in particular, in the space $Q(1,1,1)=(SU(2) \times SU(2) \times SU(2))/ (U(1) \times U(1))$ with the natural $S^1$-family of nearly parallel $G_2$-structures induced by the Sasaki-Einstein structure. In each of those cases, we obtain a family of non-trivial associative deformations.

math.DG↗
arXiv2023

Stratification of $\mathrm{SU}(r)$-character varieties of twisted Hopf links

We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $Γ_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Florentino and Lawton. In the case $r = 3$, we also describe different strata of the $\mathrm{SU}(3)$-character variety according to the semi-simple type of the representation.

math.GT↗
arXiv2023

Automorphism groups of Cayley evolution algebras

In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra.

math.RA↗
arXiv2023

On the genesis of BBP formulas

We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for $π$. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing $0$ come from. We also explain what is the observed relation between some BBP formulas for $\log 2$ and $π$, that are obtained by taking real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transalgebraic considerations, and offer a new way to obtain and to search many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants.

math.NT↗
arXiv2022

A compact $G_2$-calibrated manifold with first Betti number $b_1=1$

We construct a compact formal 7-manifold with a closed $G_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it does not admit any $G_2$-structure such that the holonomy group of the associated metric is a subgroup of $G_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed $G_2$-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over $S^2\times S^1$ with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers.

math.DG↗
arXiv2022

Geometry of $\mathrm{SU}(3)$-character varieties of torus knots

We describe the geometry of the character variety of representations of the knot group $Γ_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation, and irreducible representations, the latter of two types depending on whether the matrices have distinct eigenvalues, or one of the matrices has one eigenvalue of multiplicity $2$. We describe how the closure of each stratum meets lower strata, and use this to compute the compactly supported Euler characteristic, and to prove that the inclusion of the character variety for $\mathrm{SU}(3)$ into the character variety for $\mathrm{SL}(3,\mathbb{C})$ is a homotopy equivalence.

math.GT↗
arXiv2022

Motive of the $SL_4$-character variety of torus knots

In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$ of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem.

math.AG↗
arXiv2022

Coordinate rings of some $SL_2$-character varieties

We determine generators of the coordinate ring of $SL_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $SL_2$-character variety. For free groups $F_k$ we find transcendental generators. Finally, for the case of the 2-torus, we get an explicit equation of the $SL_2$-character variety and use the description to compute their E-polynomials.

math.AG↗