Searcharxiv⌕ Search

arXiv subjects

E. Macías-Virgós

Publications and source records attributed to E. Macías-Virgós.

At least 19 recordsLinked to original sources
arXiv2022

Homotopy invariants in small categories

Tanaka introduced a notion of Lusternik Schnirelmann category, denoted $\mathrm{ccat}\, \mathcal{C}$, of a small category $\mathcal{C}$. Among other properties, he proved an analog of Varadarajan's theorem for fibrations, relating the LS-categories of the total space, the base and the fiber. In this paper we recall the notion of homotopic distance $\mathrm{D}(F,G)$ between two functors $F,G\colon \mathcal{C} \to \mathcal{D}$, later introduced by us, which has $\mathrm{ccat} \mathcal{C}=\mathrm{D}(\mathrm{id}_{\mathcal{C}},\bullet)$ as a particular case. We consider another particular case, the distance $\mathrm{D}(p_1,p_2)$ between the two projections $p_1,p_2\colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$, which we call the categorical complexity of the small category $\mathcal{C}$. Moreover, we define the higher categorical complexity of a small category and we show that it can be characterized as a higher distance. We prove the main properties of those invariants. As a final result we prove a Varadarajan's theorem for the homotopic distance for Grothendieck bi-fibrations between small categories.

math.AT↗
arXiv2021

Homotopic distance and generalized motion planning

We prove that the homotopic distance between two maps defined on a manifold is bounded above by the sum of their subspace distances on the critical submanifol of any Morse-Bott function. This generalizes the Lusternik-Schnirelmann theorem (for Morse functions), and a similar result by Farber for the topological complexity. Analogously, we prove that, for analytic manifolds, the homotopic distance is bounded by the sum of the subspace distances on any submanifold and its cut locus. As an application, we show how navigation functions can be used to solve a generalized motion planning problem.

math.AT↗
arXiv2020

Non-unimodular transversely homogeneous foliations

We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over $S^1$.

math.DG↗
arXiv2020

Fundamental Theorems of Morse Theory on posets

We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the Morse-Pitcher inequalities in the context of finite spaces.

math.CO↗
arXiv2019

Homotopic distance between maps

We show that both Lusternik-Schnirelmann category and topological complexity are particular cases of a more general notion, that we call homotopic distance between two maps. As a consequence, several properties of those invariants can be proved in a unified way and new results arise.

math.AT↗
arXiv2019

Relative singular value decomposition and applications to LS-category

Let $Sp(n)$ be the symplectic group of quaternionic $(n\times n)$-matrices. For any $1\leq k\leq n$, an element $A$ of $Sp(n)$ can be decomposed in $A= \begin{bmatrix} α&T\cr β&P \end{bmatrix}$ with $P$ a $(k\times k)$-matrix. In this work, starting from a singular value decomposition of $P$, we obtain what we call a relative singular value decomposition of $A$. This feature is well adapted for the study of the quaternionic Stiefel manifold $X_{n,k}$, and we apply it to the determination of the Lusternik-Schnirelmann category of $Sp(k)$ in $X_{2k-j,k}$, for $j= 0,\,1,\,2$

math.AT↗
arXiv2019

Simplicial fibrations

We undertake a systematic study of the notion of fibration in the setting of abstract simplicial complexes, where the concept of `homotopy' has been replaced by that of `contiguity'. Then a fibration will be a simplicial map satisfying the `contiguity lifting property'. This definition turns out to be equivalent to a known notion introduced by G. Minian, established in terms of a cylinder construction $K \times I_m$. This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration have the same strong homotopy type, a notion that has been recently introduced by Barmak and Minian; any fibration with a strongly collapsible base is fibrewise trivial; and some other ones. We introduce the concept of `simplicial finite-fibration', that is, a map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration $PK \to K\times K$ is a finite-fibration, where PK is the space of Moore paths introduced by M. Grandis. This important result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we introduce a definition of `Švarc genus' of a simplicial map and, and using the properties stated before, we are able to compare the Švarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers. Finally, another key result is a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber.

math.AT↗
arXiv2019

Homotopic distance between functors

We introduce a notion of categorical homotopic distance between functors by adapting the notion of homotopic distance in topological spaces, recently defined by the authors to the context of small categories. Moreover, this notion generalizes the work on categorical LS-category of small categories by Tanaka.

math.AT↗
arXiv2018

Euler calculus on spaces homeomorphic to definable sets and some applications

We show that integration with respect to the Euler-Poincaré characteristic can be extended from the setting of definable sets to the setting of topological spaces homeomorphic to definable sets. We use that extension to generalize a result regarding sensor networks due to Ghrist and Baryshnikov, in order to make it more flexible in applications. Finally, we obtain both an extension and a combinatorial proof of a classical result about the Euler-Poincaré characteristic of fiber bundles.

math.AT↗
arXiv2017

Discrete Topological complexity

We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex by means of the concept of contiguous simplicial maps. We study the links of this new invariant with those of simplicial and topological LS-category.

math.AT↗
arXiv2015

Lusternik-Schnirelmann category of simplicial complexes and finite spaces

In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of geometric category for simplicial complexes. We prove that the maximum value over the homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of category are obtained in the setting of finite topological spaces.

math.AT↗
arXiv2013

Height functions on compact symmetric spaces

We consider height functions on symmetric spaces $M\cong G/K$ embedded in the associated matrix Lie group $G$. In particular we study the relationship between the critical sets of the height function on $G$ and its restriction to $M$. Also we prove that the gradient flow on $M$ can be integrated by means of a generalized Cayley transform. This allows to obtain explicit local charts for the critical submanifolds. Finally, we discuss how to reduce the generic case to a height function whose ground hyperplane is orhogonal to a real diagonal matrix. This result requires to prove the existence of a polar decomposition adapted to the automorphism defining $M$. Detailed examples are given.

math.DG↗
arXiv2012

A topological approach to left eigenvalues of quaternionic matrices

It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted by computing the topological degree of a characteristic map associated to the matrix and discussing the rank of the differential. The same techniques are extended to $3\times 3$ matrices, which are still lacking a complete classification.

math.RA↗
arXiv2010

Characteristic functions and Hamilton-Cayley theorem for left eigenvalues of quaternionic matrices

We introduce the notion of characteristic function of a quaternionic matrix, whose roots are the left eigenvalues. We prove that for all $2\times 2$ matrices and for $3\times 3$ matrices having some zero entry outside the diagonal there is a characteristic function which is a polynomial. For the other $3\times 3$ matrices the characteristic function is a rational function with one point of discontinuity. We prove that Hamilton-Cayley theorem holds in all cases.

math.RA↗
arXiv2009

Symplectic matrices with predetermined left eigenvalues

We prove that given four arbitrary quaternion numbers of norm 1 there always exists a $2\times 2$ symplectic matrix for which those numbers are left eigenvalues. The proof is constructive. An application to the LS category of Lie groups is given.

math.RA↗