arXiv subjects

José Antonio Arciniega-Nevárez

Publications and source records attributed to José Antonio Arciniega-Nevárez.

5 recordsLinked to original sources

arXiv2026

Bicomplex Characteristic Classes

We introduce bicomplex fiber bundles and develop a framework for bicomplex characteristic classes and almost bicomplex structures on smooth manifolds. In the first part, we present the basic definitions and properties of these bundles, including their fundamental idempotent decomposition and the correspondence between their isomorphism classes and homotopy classes of maps into the bicomplex infinite Grassmannian. We then define bicomplex characteristic classes and show that, in general, they are not determined by the Chern classes of the underlying complex bundle. We also construct the bicomplex Chern character and discuss its algebraic properties. Finally, we apply the resulting theory to almost bicomplex manifolds, providing examples and obstructions to the existence of such structures.

math.AG↗

arXiv2026

Operad of posets 101: The Wixárika posets

We study classes of objects whose combinatorics are closely related to those of posets. The framework of operads and operad algebras allows us to make this relationship precise and provides tools for a deeper understanding of their combinatorial structure. In this note, we present a nontrivial example of a suboperad of the operad of posets, called Wixárika posets, together with its associated algebras. This example is sufficiently rich to exhibit key structural features of the theory, while remaining accessible and avoiding unnecessary technicalities.

math.CO↗

arXiv2023

Cheeger-Chern-Simons classes of representations of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and the spectrum of rational double point singularities

Let $L$ be a compact oriented $3$-manifold and $ρ\colonπ_1(L)\to \mathrm{GL}(n,\mathbb{C})$ a representation. Evaluating the Cheeger-Chern-Simons class $\widehat{c}_{ρ,k}\in H^{2k-1}(L;\mathbb{C}/\mathbb{Z})$ of $ρ$ at $ν\in H_{2k-1}(L;\mathbb{Z})$ we get characteristic numbers that we call the $k$-th CCS-numbers of $ρ$. We prove that if $ρ$ is a topologically trivial representation, the 2-nd CCS-number $\widehat{c}_{ρ,2}([L])$ of the fundamental class $[L]$ of $L$ is given by the invariant $\tildeξ_ρ(D)$ of the Dirac operator $D$ of $L$ twisted by $ρ$ defined by Atiyah, Patodi and Singer. If $L$ is a rational homology sphere, we also give a formula for $\widehat{c}_{ρ,2}([L])$ of any representation $ρ$ in terms of $\tildeξ(D)$. Given a topologically trivial representation $ρ\colonπ_1(L)\to\mathrm{GL}(n,\mathbb{C})$ we construct an element $\langle L,ρ\rangle$ in the $3$-rd algebraic K-theory group $K_3(\mathbb{C})$ of the complex numbers. For a finite subgroup $Γ$ of $\mathrm{SU(2)}$ and its irreducible representations, we compute the 1-st and 2-nd CCS-numbers. With this, we recover the spectrum of all rational double point singularities. Motivated by this result, we define the topological spectrum of rational surface singularities and Gorenstein singularities. Given a normal surface singularity $(X,x)$ with link a rational homology sphere $L$, we show how to compute the invariant $\tildeξ_ρ(D)$ for the Dirac operator of $L$ using either a resolution or a smoothing of $(X,x)$.

math.DG↗

arXiv2020

Relative group (co)homology theories with coefficients and the comparison homomorphism

Let $G$ be a group, let $H$ be a subgroup of $G$ and let $\Or(G)$ be the orbit category. In this paper we extend the definition of the relative group (co)homology theories of the pair $(G,H)$ defined by Adamson and Takasu to have coefficients in an $\Or(G)$-module. There is a canonical comparison homomorphism defined by Cisneros-Molina and Arciniega-Nevárez from Takasu's theory to Adamson's one. We give a necessary and sufficient condition on the subgroup $H$ for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson (co)homology. Finally, we provide some examples of explicit computations for the comparison homomorphism.

math.KT↗

arXiv2018

Invariants of hyperbolic 3-manifolds in relative group homology

Let $M$ be a complete oriented hyperbolic $3$--manifold of finite volume. Using classifying spaces for families of subgroups we construct a class $β_P(M)$ in the Adamson relative homology group $H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$, where $\bar{P}$ is the subgroup of parabolic transformations which fix $\infty$ in the Riemann sphere. We also prove that the classes $F(M)$ in the Takasu relative homology groups $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$ constructed by Zickert, which are not well-defined and depend of a choice of decorations by horospheres, are all mapped to $β_P(M)$ via a canonical comparison homomorphism $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\to H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$. To do this, we simplify the construction of the classes $F(M)$ using a simpler complex which computes $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$, getting a simple simplicial formula for $F(M)$, which in turn gives a simpler and more efficient formula to compute the volume and Chern--Simons invariant than the one given by Zickert. The constructions can be extended for any boundary-parabolic $PSL_2(\mathbb{C})$-representation.

math.GT↗