A theoretical framework for proper homotopy theory

Quintero, A.; Domı́nguez, Elena

doi:10.1017/s0305004100068754

Cited by 16 publications

(19 citation statements)

References 21 publications

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“…Hence card(^(X)) < p-cat(X). of a p-map / is defined in the natural way (see [2] for details). If /= r, C p r turns to be the proper suspension Σ p X of X.…”

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confidence: 99%

Lusternik-Schnirelmann invariants in proper homotopy theory

Domı́nguez¹,

Márquez²,

Quintero³

1992

Pacific J. Math.

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We introduce and study proper homotopy invariants of the Lusternik-Schnirelmann type, p-cat (-), p-Cat(-), and cat e(-) in the category of Γ2-locally compact spaces and proper maps. As an application, R Here we present the definition and the basic properties of a new numerical topological invariant for Γ 2 -locally compact spaces which agrees with the notion of L-S category for ^-compact spaces. This invariant, denoted p-cat(X), is called the proper L-S category of X and turns out to be a proper homotopy invariant of X. Hence, p-cat(X) is a finer invariant than cat(X).In [10] several generalizations of L-S category are suggested. More explicitly, a general notion of L-S si -category with respect to a class si of spaces has been developed by Puppe and Clapp in [6]. Our work shares some common points with [6] but does not fit into the notion of L-S si -category since we entirely deal with proper maps instead of ordinary continuous maps.Another generalization of L-S category has been given in [1], where L-S category for pro-objects in pro-c%/* is defined. This idea is related to proper L-S category by the Edwards-Hastings embedding (see [8]) which provides a close link between proper homotopy theory and homotopy in

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“…Hence card(^(X)) < p-cat(X). of a p-map / is defined in the natural way (see [2] for details). If /= r, C p r turns to be the proper suspension Σ p X of X.…”

mentioning

confidence: 99%

Lusternik-Schnirelmann invariants in proper homotopy theory

Domı́nguez¹,

Márquez²,

Quintero³

1992

Pacific J. Math.

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“…The composite p 1 : P R + (X, α)→X I d 1 → X, is the corresponding path fibration. Then, G n (X, α) is the join of G n−1 (X, α) p n−1 → X α ← R + , and p n is the induced whisker map:…”

Section: Lemma 3·2 Let (X α) Be a Normal Well-based Exterior Spacementioning

confidence: 99%

A Whitehead–Ganea approach for proper Lusternik–Schnirelmann category

Calcines

García-Díaz

Murillo-Mas

2007

Math. Proc. Camb. Phil. Soc.

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“…In fact, the category P is a cofibration category in the sense of Baues ([4] and [3]). Moreover in P the role of the base point is played by the half-line R + = [0, ∞) since [X, R + ] p = { * } is a one-point set [8].…”

Section: Preliminaries Recall That a Continuous Mapmentioning

confidence: 99%

“…We can do that since the crucial points in [17] are general position in pl-manifolds as well as some features of the ordinary homotopy category which lie in the cofibration part of homotopy theory. In the proper category we shall use the (proper) general position theorem of [12] as well as the structure of cofibration category in the sense of Baues of the proper category; see [3].…”

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Embedding proper homotopy types

et al. 2003

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Abstract. We show that the proper homotopy type of any properly c-connected locally finite n-dimensional CW-complex is represented by a closed polyhedron in R 2n−c (Theorem I). The case n − c ≥ 3 is a special case of a general proper homotopy embedding theorem (Theorem II). For n − c ≤ 2 we need some basic properties of "proper" algebraic topology which are summarized in Appendices A and B. The results of this paper are the proper analogues of classical results by Stallings [17] and Wall [20] for finite CW-complexes; see also Dranišnikov and Repovš [7].

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A theoretical framework for proper homotopy theory

Abstract: Following the techniques of ordinary homotopy theory, a theoretical treatment of proper homotopy theory, including the known proper homotopy groups, is provided within Baues's theory of cofibration categories.

Cited by 16 publications

References 21 publications

Lusternik-Schnirelmann invariants in proper homotopy theory

Lusternik-Schnirelmann invariants in proper homotopy theory

A Whitehead–Ganea approach for proper Lusternik–Schnirelmann category

Embedding proper homotopy types

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