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Jesús González

Publications and source records attributed to Jesús González.

At least 37 records · Page 2Linked to original sources
arXiv2020

Equivariant Nerve Lemma, simplicial difference, and models for configuration spaces on simplicial complexes

Wiltshire-Gordon has introduced a homotopy model for ordered configuration spaces on a given simplicial complex. That author asserts that, after a suitable subdivision, his model also works for unordered configuration spaces. We supply details justifying Wiltshire-Gordon's assertion and, more importantly, uncover the equivariant properties of his more-general simplicial-difference model for the complement of a subcomplex inside a larger complex. This is achieved by proving an equivariant version of the Nerve Lemma. In addition, in the case of configuration spaces, we show that a slight variation of the model has better properties: it is regular and sits inside the configuration space as a strong and equivariant deformation retract. Our variant for the configuration-space model comes from a comparison, in the equivariant setting, between Wiltshire's simplicial difference and a well known model for the complement of a full subcomplex on a simplicial complex.

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arXiv2020

Linear motion planning with controlled collisions and pure planar braids

We compute the Lusternik-Schnirelmann category (LS-cat) and the higher topological complexity ($TC_s$, $s\geq2$) of the "no-$k$-equal" configuration space Conf$_k(\mathbb{R},n)$. This yields (with $k=3$) the LS-cat and the higher topological complexity of Khovanov's group PP$_n$ of pure planar braids on $n$ strands, which is an $\mathbb{R}$-analogue of Artin's classical pure braid group on $n$ strands. Our methods can be used to describe optimal motion planners for PP$_n$ provided $n$ is small.

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arXiv2020

Effective topological complexity of orientable-surface groups

We use rewriting systems to spell out cup-products in the (twisted) cohomology groups of a product of surface groups. This allows us to detect a non-trivial obstruction bounding from below the effective topological complexity of an orientable surface with respect to its antipodal involution. Our estimates are at most one unit from being optimal, and are closely related to the (regular) topological complexity of non-orientable surfaces.

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arXiv2019

Sequential collision-free optimal motion planning algorithms in punctured Euclidean spaces

In robotics, a topological theory of motion planning was initiated by M. Farber. The multitasking motion planning problem is new and its theoretical part via topological complexity has hardly been developed, but the concrete implementations are still non-existent, and in fact this work takes the first step in this last direction (producing explicit algorithms.) We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multitasking version of the algorithms.

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arXiv2019

Farley-Sabalka's Morse-theory model and the higher topological complexity of ordered configuration spaces on trees

Using the ordered analogue of Farley-Sabalka's discrete gradient field on the configuration space of a graph, we unravel a levelwise behavior of the generators of the pure braid group on a tree. This allows us to generalize Farber's equivariant description of the homotopy type of the configuration space on a tree on two particles. The results are applied to the calculation of all the higher topological complexities of ordered configuration spaces on trees on any number of particles.

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arXiv2017

Hopf Invariants for sectional category with applications to topological robotics

We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied in the study of Farber's topological complexity for 2-cell complexes, as well as in the construction of a counter-example to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category.

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arXiv2017

Symmetric bi-skew maps and symmetrized motion planning in projective spaces

This work is motivated by the question of whether there are spaces $X$ for which the Farber-Grant symmetric topological complexity $TC^S(X)$ differs from the Basabe-González-Rudyak-Tamaki symmetric topological complexity $TC^Σ(X)$. It is known that, for a projective space $RP^m$, $TC^S(RP^m)$ captures, with a few potentially exceptional cases, the Euclidean embedding dimension of $RP^m$. We now show that, for all $m\geq1$, $TC^Σ(RP^m)$ is characterized as the smallest positive integer $n$ for which there is a symmetric $\mathbb{Z}_2$-biequivariant map $S^m\times S^m\to S^n$ with a "monoidal" behavior on the diagonal. This result thus lies at the core of the efforts in the 1970's to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both $TC$ numbers in the case of $RP^{2^e}$ for $e\geq1$. In particular, this leaves the torus $S^1\times S^1$ as the only closed surface whose symmetric (symmetrized) $TC^S$ ($TC^Σ$) -invariant is currently unknown.

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arXiv2017

Simplicial Complexity: piecewise linear motion planning in robotics

Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexity of the realization $\|K\|$. Our approach is well suited for designing and implementing algorithms that search for optimal motion planners for autonomous systems in real-life applications.

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arXiv2016

ALMA and RATIR observations of GRB131030A

We report on the first open-use based Atacama Large Millimeter/submm Array (ALMA) 345-GHz observation for the late afterglow phase of GRB131030A. The ALMA observation constrained a deep limit at 17.1 d for the afterglow and host galaxy. We also identified a faint submillimeter source (ALMAJ2300-0522) near the GRB131030A position. The deep limit at 345 GHz and multifrequency observations obtained using {\it Swift} and RATIR yielded forward shock modeling with a two-dimensional relativistic hydrodynamic jet simulation and described X-ray excess in the afterglow. The excess was inconsistent with the synchrotron self-inverse Compton radiation from the forward shock. The host galaxy of GRB131030A and optical counterpart of ALMAJ2300-0522 were also identified in the SUBARU image. Based on the deep ALMA limit for the host galaxy, the 3-$σ$ upper limits of IR luminosity and the star formation rate (SFR) is estimated as $L_{IR}<1.11\times10^{11} L_{\odot}$ and SFR$<18.7$ ($M_{\odot}$~yr$^{-1}$), respectively. Although the separation angle from the burst location (3.5 arcsec) was rather large, ALMAJ2300-0522 may be one component of the GRB131030A host galaxy, according to previous host galaxy cases.

astro-ph.HE↗
arXiv2016

The stability of the higher topological complexity of real projective spaces: an approach to their immersion dimension

The $s$-th higher topological complexity of a space $X$, $TC_s(X)$, can be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when $X=RP^m$, the real projective space of dimension $m.$ In particular, we describe a number $r(m)$, which depends on the structure of zeros and ones in the binary expansion of $m$, and with the property that $TC_s(RP^m)$ is given by $sm$ with an error of at most one provided $s \geq r(m)$ and $m \not\equiv 3 \bmod 4$ (the error vanishes for even $m$). The latter fact appears to be closely related to the estimation of the Euclidean immersion dimension of $RP^m$. We illustrate the phenomenon in the case $m=3 \cdot 2^a$.

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arXiv2016

SDSS IV MaNGA: The global and local stellar mass assemby histories of galaxies

By means of the fossil record method implemented through Pipe3D, we reconstruct the global and radial stellar mass growth histories (MGHs) of an unprecedentedly large sample of galaxies, ranging from dwarf to giant objects, from the "Mapping Nearby Galaxies at the Apache Point Observatory" survey. We confirm that the main driver of the global MGHs is mass, with more massive galaxies assembling their masses earlier (downsizing), though for a given mass, the global MGHs segregate by color, specific star formation rate (sSFR), and morphological type. From the inferred radial mean MGHs, we find that at the late evolutionary stages (or for fractions of assembled mass larger than ~ 80%), the innermost regions formed stars on average earlier than the outermost ones (inside-out). At earlier epochs, when the age resolution of the method becomes poor, the mass assembly seems to be spatially homogeneous or even in the outside-in mode, specially for the red/quiescent/early-type galaxies. The innermost MGHs are in general more regular (less scatter around the mean) than the outermost ones. For dwarf and low-mass galaxies, we do not find evidence of an outside-in formation mode; instead their radial MGHs are very diverse most of the time, with periods of outside- in and inside-out modes (or strong radial migration), suggesting this an episodic SF history. Blue/star-forming/late-type galaxies present on average a significantly more pronounced inside-out formation mode than red/quiescent/early-type galaxies, independently of mass. We discuss our results in the light of the processes of galaxy formation, quenching, and radial migration. We discuss also on the uncertainties and biases of the fossil record method and how they could affect our results.

astro-ph.GA↗
arXiv2016

Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

Let $X$ be a two-cell complex with attaching map $α\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1<q\leq3(p-1)$ under fairly mild conditions by making use of the Hopf invariant techniques recently developed by the authors in their study of the sectional category of arbitrary maps.

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arXiv2016

Pairwise disjoint maximal cliques in random graphs and sequential motion planning on random right angled Artin groups

The clique number of a random graph in the Erdos-Renyi model G(n,p) yields a random variable which is known to be asymptotically (as n tends to infinity) almost surely within one of an explicit logarithmic (on n) function r(n,p). We extend this fact by showing that random graphs have, asymptotically almost surely, arbitrarily many pairwise disjoint complete subgraphs with as many vertices as r(n,p). The result is motivated by and applied to the sequential motion planning problem on random right angled Artin groups. Indeed, we give an asymptotical description of all the higher topological complexities of Eilenberg-MacLane spaces associated to random graph groups.

math.CO↗
arXiv2015

Motion planning in real flag manifolds

Starting from Borel's description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring for semi complete flag manifolds $F_{k,m}:=F(1,\ldots,1,m)$ where $1$ is repeated $k$ times. The information is used in order to estimate Farber's topological complexity of these spaces when $m$ approaches (from below) a 2-power. In particular, we get almost sharp estimates for $F_{2,2^e-1}$ which resemble the known situation for the real projective spaces $F_{1,2^e}$. Our results indicate that the agreement between the topological complexity and the immersion dimension of real projective spaces no longer holds for other flag manifolds. More interestingly, we also get corresponding results for the $s$-th (higher) topological complexity of these spaces. Actually, we prove the surprising fact that, as $s$ increases, the estimates become stronger. Indeed, we get several full computations of the higher motion planning problem of these manifolds. This property is also shown to hold for surfaces: we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not). A homotopy-obstruction explanation is included for the phenomenon of having a cohomologically accessible higher topological complexity even when the regular topological complexity is not so accessible.

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arXiv2015

Induced nuclear activity in galaxy pairs with different morphologies (E+E), (E+S) and (S+S)

We analysed 385 galactic spectra from the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7) that belong to the catalog of isolated pairs of galaxies by Karachentsev. The spectra corresponds to physical pairs of galaxies as defined by V $\leq$ 1200 Km/s and a pair separation $\leq$ 100 kpc. We search for the incidence of nuclear activity, both thermal (star-forming) and non-thermal -Active Galactic Nuclei (AGN). After a careful extraction of the nuclear spectra, we use diagnostic diagrams and find that the incidence of AGN activity is 48 \% in the paired galaxies with emission lines and 40\% for the total sample (as compared to $\sim$ 43 \% and 41\% respectively in a sample of isolated galaxies). These results remain after dissecting the effects of morphological type and galactic stellar mass (with only a small, non significant, enhancement of the AGN fraction in pairs of objects). These results suggest that weak interactions are not necessary or sufficient to trigger low-luminosity AGN. Since the fraction of AGN is predominant in early type spiral galaxies, we conclude that the role of a bulge, and a large gas reservoir are both essential for the triggering of nuclear activity. The most striking result is that type 1 galaxies are almost absent from the AGN sample. This result is in conflict with the Unified Model, and suggests that high accretion rates are essential to form the Broad Line Region in active galaxies.

astro-ph.GA↗
arXiv2015

The cohomology ring away from 2 of configuration spaces on real projective spaces

Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space F(m,k) of k ordered points in the m-dimensional real projective space. The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2^k-fold covering of F(m,k). This implies that, for odd m, the Leray spectral sequence for the inclusion of F(m,k) in the k-fold Cartesian self power of the m-dimensional real projective space collapses after its first non-trivial differential, just as it does when the projective space is replaced by a complex projective variety. The method also allows us to handle the R-cohomology ring of the configuration space of k ordered points in a punctured real projective space. Lastly, we compute the Lusternik-Schnirelmann category and all of the higher topological complexities of some of the auxiliary orbit configuration spaces.

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