arXiv subjects

J. M. García-Calcines

Publications and source records attributed to J. M. García-Calcines.

6 recordsLinked to original sources

arXiv2026

New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance

In this paper, we establish several new bounds for the distributional sectional category ($\mathrm{dsecat}$). We first prove Jauhari's conjecture, thereby establishing a cohomological lower bound for $\mathrm{dsecat}$ with arbitrary coefficients. We then obtain an analogous lower bound with rational coefficients by constructing a natural splitting of the map induced on cohomology by the diagonal inclusion into symmetric powers. As applications, we provide several computations of the distributional sectional category. We also establish multiplicative product and composition inequalities for $\mathrm{dsecat}$. Finally, we apply these results to the distributional homotopic distance recently introduced by Jauhari and Oprea, giving an equivalent formulation in terms of distributed homotopies and deriving new fibration and multiplicative triangle inequalities.

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arXiv2025

Equivariant homotopic distance

We introduce and study the notion of \emph{equivariant homotopic distance} $D_G(f,g)$ between $G$-maps $f,g \colon X \to Y$. We show that the equivariant Lusternik-Schnirelmann category and the equivariant topological complexity are particular cases of this notion. This invariant also connects naturally with the equivariant sectional category. What makes $D_G$ distinctive, however, is that it provides a flexible framework centered on pairs of maps, within which one can derive results that are not immediate from the general setting. In particular, we establish its basic properties, including homotopy invariance and a categorical proof of the triangle inequality valid in the equivariant context. We also obtain cohomological and dimension-connectivity bounds, and analyze structural applications to Hopf $G$-spaces and equivariant fibrations.

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arXiv2025

Parametrized homotopic distance

We introduce the concept of parametrized homotopic distance, extending the classical notion of homotopic distance to the fibrewise setting. We establish its correspondence with the fibrewise sectional category of a specific fibrewise fibration and derive cohomological lower bounds and connectivity upper bounds under mild conditions. We also analyze the behavior of parametrized homotopic distance under compositions and products of fibrewise maps, along with its interaction with the triangle inequality. We establish several sufficient conditions for fibrewise $H$-spaces to admit a fibrewise division map and prove that their parametrized topological complexity equals their fibrewise unpointed Lusternik-Schnirelman category, extending Lupton and Scherer's theorem to the fibrewise setting. Additionally, we give sharp estimates for the parametrized topological complexity of a class fibrewise $H$-spaces which arises as sphere bundles with fibre $S^7$. Furthermore, we estimate the parametrized homotopic distance of fibre-preserving, fibrewise maps between fibrewise fibrations, in terms of the parametrized homotopic distance of the induced fibrewise maps between individual fibres, as well as the fibrewise unpointed Lusternik-Schnirelman category of the base space. Finally, we define and study a pointed version of parametrized homotopic distance, establishing cohomological bounds and identifying key conditions for its equivalence with the unpointed version, thus providing a finer classification of fibrewise homotopy invariants.

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arXiv2024

Generalized topological complexity and its monoidal version

In the context of the Lusternik-Schnirelmann category, researcher T. Srinivasan demonstrated that when the space under consideration is an absolute neighborhood retract, its category can be realized through arbitrary subsets, not necessarily open ones. The primary aim of this survey is to illustrate how this result has been extended to the case of topological complexity and its monoidal version, along with some of its most significant implications.

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arXiv2014

Regions of attraction, limits and end points of an exterior discrete semi-flow

An exterior space is a topological space provided with a quasi-filter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow. On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (Steenrod, \v Cech, Brown-Grossman). On the other hand, for a discrete semi-flow one can analyze fixed points, periodic points, omega-limits, et cetera. In this paper, we see that a discrete semi-flow can be provided with the canonical structure of an exterior space given by the family of right-absorbing open subsets that can be used to study the relation between limits and periodic points and connections between bar-limits and omega-limits. The different notions of end point can be used to decompose the region of attraction of an exterior discrete semi-flow as a disjoint union of basins of end points.

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arXiv2014

Brown representability for exterior cohomology and cohomology with compact supports

It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem reachable. However, by proving such a theorem for the so called exterior cohomology in the complete and cocomplete exterior category, we show that the $n$-th cohomology with compact supports of a given countable, locally finite, finite dimensional relative CW-complex $(X,\mathbb{R}_+)$ is naturally identified with the set $[X,K_n]^{\mathbb{R}_+}$ of exterior based homotopy classes from a "classifying space" $K_n$. We also show that this space has the exterior homotopy type of the exterior Eilenberg-MacLane space for Brown-Grossman homotopy groups of type $(R^\infty,n)$, $R$ being the fixed coefficient ring.

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