Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

Calcines, José Manuel García; García-Pinillos, Mónica; Paricio, Luis Javier Hernández

doi:10.1023/b:apcs.0000031087.83413.a4

Cited by 16 publications

(14 citation statements)

References 19 publications

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“…Several analogues of this type of fundamental group have been given in different contexts, for example, in proper homotopy theory by Porter [27] and by Hughes and Ranicki [40], in homotopy theory of prospaces by Edwards-Hastings [16], in (strong) shape theory by Quigley [25], Cathey [41] and for exterior spaces by Calcines-Hernández-Pinillos [28,29]. The reason to include in the name 'Steenrod' is the existence of a Theorem of Hurewicz type between these groups and the Steenrod homology groups.…”

Section: The Steenrod-quigley Fundamental Groupmentioning

confidence: 99%

Fundamental groups and finite sheeted coverings

Paricio

Matijević

2010

Journal of Pure and Applied Algebra

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Communicated by E.M. Friedlander MSC: 55P57 55Q52 55Q70 55P55 55U40 55Q07 55R65 a b s t r a c tIt is well known that for a connected locally path-connected semi-locally 1-connected space X , there exists a bi-unique correspondence between the pointed d-fold connected coverings and the transitive representations of the fundamental group of X in the symmetric group Σ d of degree d.The classification problem becomes more difficult if X is a more general space, particularly if X is not locally connected. In attempt to solve the problem for general spaces, several notions of coverings have been introduced, for example, those given by Lubkin or by Fox. On the other hand, different notions of 'fundamental group' have appeared in the mathematical literature, for instance, the Brown-Grossman-Quigley fundamental group, the Čech-Borsuk fundamental group, the Steenrod-Quigley fundamental group, the fundamental profinite group or the fundamental localic group.The main result of this paper determines different 'fundamental groups' that can be used to classify pointed finite sheeted connected coverings of a given space X depending on topological properties of X .

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Section: The Steenrod-quigley Fundamental Groupmentioning

confidence: 99%

Fundamental groups and finite sheeted coverings

Paricio

Matijević

2010

Journal of Pure and Applied Algebra

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“…An illustrative example is given by the fact that the category P of topological spaces and proper maps is an I-category [3], and thus it is also a cofibration category, while even the definition of what a proper fibration should be presents some issues [2,7]. However, P is embedded in the category E of exterior spaces which is complete and cocomplete and it has model category structures [11,12,8] closely related to the classical ones on topological spaces (see §1 for notation and terminology). From this, the concept of proper fibration becomes clear as exterior fibration in the proper category.…”

Section: Introductionmentioning

confidence: 99%

Classification of exterior and proper fibrations

Calcines

García-Díaz

Murillo

2020

Proc. Amer. Math. Soc.

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“…The category of exterior spaces has been provided with a well developed homotopy theory ( [12,[16][17][18]21]). The study of the exterior and proper homotopy invariants has proved to be useful in the study of non-compact manifolds ( [8,30]), the study of the shape of some compact spaces ( [22]), the L-S proper category ( [14,15]), et cetera.…”

Section: Introductionmentioning

confidence: 99%

“…An answer to this problem is given by the notion of exterior space. The new category of exterior spaces and maps is complete and cocomplete and contains as a full subcategory the category of spaces and proper maps, see [16,17]. We refer to [9,11,12,21,22] for further properties and applications of exterior homotopy, and to [26] for a survey of proper homotopy.…”

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confidence: 99%

Omega limits, prolongational limits and almost periodic points of a continuous flow via exterior spaces

Calcines

Paricio

Rodríguez

2017

Glas. Mat. Ser. III

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Abstract. In this paper we analyse some applications of the category of exterior spaces to the study of dynamical systems (flows). The limit space and end space of an exterior space are used to construct different types of limit spaces and end spaces of a dynamical system. In this work we analyse the relationships between the notions and constructions given by the exterior structures of a continuous flow and the more usual notions of omega-limits, first prolongational limits and several types of almost periodic points (Poisson-stable points, non-wandering points) of a flow.

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Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

Cited by 16 publications

References 19 publications

Fundamental groups and finite sheeted coverings

Fundamental groups and finite sheeted coverings

Classification of exterior and proper fibrations

Omega limits, prolongational limits and almost periodic points of a continuous flow via exterior spaces

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