The proper L-S category of Whitehead manifolds

It is shown that the proper L-S category of an eventually end-irreducible, R²-irreducible Whitehead 3-manifold is 4. For this we prove, in the category of germs at infinity of proper maps, a partial analogue of the characterization by Eilenberg and Ganea of the L-S category of an aspherical space.     

The coshape invariant and continuous extensions of functors

The coshape invariant and continuous extensions of group-valued covariant and contravariant functors, defined on the category of pairs of spaces with the homotopy type of a pair of finite CW-complexes, are constructed.     

Non-Abelian extensions and homotopies

Let n+k>n2. We consider algebraic models of the homotopy category of CW-complexes with cells only in dimension n and n+k. The algebraic model category is described in terms of non-Abelian extensions in Ext².     

Lusternik-Schnirelmann category of a sphere-bundle over a sphere

We determine the Lusternik-Schnirelmann (L-S) category of a total space of a sphere-bundle over a sphere in terms of primary homotopy invariants of its characteristic map, and thus providing a complete answer to Ganea's Problem 4. As a result, we obtain a necessary and sufficient condition for a total space N to have the same L-S category as its ‘once punctured submanifold’ N\{P},P∈N. Also, necessary and sufficient conditions for a total space M to satisfy Ganea's conjecture are described.     

L-S category of quaternionic Stiefel manifolds

By calculating certain generalized cohomology theory, lower bounds for the L-S category of quaternionic Stiefel manifolds are given.     

Exponential homeomorphisms in the category of topological spaces with base point

Topologies on base point preserving function spaces are studied making use of the Brown-Booth-Tillotson C-smash product and the topologies on function spaces defined by the classes C of exponentiable spaces. Some conditions on the classes C are obtained for exponential bijections and exponential homeomorphisms in the category of topological spaces with base point. Conditions for exponential homeomorphisms are obtained for CW-complexes whose cellular structure implies precise results.     

Group-like structures and the whitehead product in pro-homotopy and shape

The notion of ‘H-space’ is of considerable importance in the homotopy theory of CW-complexes. This paper studies a similar notion in the framework of pro-homotopy and shape theories. This is achieved by following the general plan set forth by Eckmann and Hilton. Examples of shape H-space are also given; it is observed that every compact connected topological monoid is a shape H-space. The Whitehead product is defined and studied in the pro-homotopy and shape categories; and, it is shown that this Whitehead product vanishes on an H-object in pro-homotopy. These results are the natural extension of some well-known classical results in the homotopy theory of CW-complexes.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

Functors on the category of quasi-fibrations

We consider the following questions: when can we extend a continuous endofunctor on Top the category of topological spaces to a fibrewise continuous endofunctor on Top(2) the category of continuous maps? If this is true, does such fibrewise continuous endofunctor preserve fibrations? In this paper, we define Fib the topological category of cell-wise trivial fibre spaces over polyhedra and show that any continuous endofunctor on Top induces a fibrewise continuous endofunctor on Fib preserving the class of quasi-fibrations.     

Automate parallèle à homotopie près (II)

One proves that the category of globular CW-complexes up to dihomotopy is equivalent to the category of flows up to weak dihomotopy. This theorem generalizes the classical theorem which states that the category of CW-complexes up to homotopy is equivalent to the category of topological spaces up to weak homotopy. To cite this article: P. Gaucher, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A generalization of Quillen's small object argument

We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several important examples of model categories which were proven to be non-cofibrantly generated [J. Adámek, H. Herrlich, J. Rosický, W. Tholen, Weak factorization systems and topological functors, Appl. Categ. Structures 10 (3) (2002) 237-249 [2]; Papers in honour of the seventieth birthday of Professor Heinrich Kleisli (Fribourg, 2000); B. Chorny, The model category of maps of spaces is not cofibrantly generated, Proc. Amer. Math. Soc. 131 (2003) 2255-2259; J.D. Christensen, M. Hovey, Quillen model structures for relative homological algebra, Math. Proc. Cambridge Philos. Soc. 133 (2) (2002) 261-293; D.C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805-2841]. Our current approach allows for construction of functorial factorizations and localizations in the equivariant model structures on diagrams of spaces [E.D. Farjoun, Homotopy theories for diagrams of spaces, Proc. Amer. Math. Soc. 101 (1987) 181-189] and diagrams of chain complexes. We also formulate a non-functorial version of the argument, which applies in two different model structures on the category of pro-spaces [D.A. Edwards, H.M. Hastings, ?ech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Mathematics, vol. 542, Springer, Berlin, 1976; D.C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc. 353 (2001) 2805-2841].The examples above suggest a natural extension of the framework of cofibrantly generated model categories. We introduce the concept of a class-cofibrantly generated model category, which is a model category generated by classes of cofibrations and trivial cofibrations satisfying some reasonable assumptions.     

A geometric decomposition of spaces into cells of different types II: Homology theory

We develop the homology theory of CW(A)-complexes, generalizing the classical cellular homology theory for CW-complexes. A CW(A)-complex is a topological space which is built up out of cells of a certain core A.     

Truncated resolution model structures

Using the dual of Bousfield-Friedlander localization, we colocalize resolution model structures on cosimplicial objects over a left proper model category to get truncated resolution model structures. These are useful for studying realization and moduli problems in algebraic topology.     

A Reidemeister trace for fibred maps

A Reidemeister trace for fibred maps is defined as the alternating sum of suitable (elementary) traces for linear morphisms of fibred cellular free modules with local coefficients. This invariant extends in a natural way the classical construction of the generalized Lefschetz number??Reidemeister trace??to the category of fibred CW-complexes.     

Strong cofibrations and fibrations in enriched categories

We define strong cofibrations and fibrations in suitably enriched categories using the relative homotopy extension resp. lifting property. We prove a general pairing result, which for topological spaces specializes to the well-known pushout-product theorem for cofibrations. Strong cofibrations and fibrations give rise to cofibration and fibration categories in the sense of homotopical algebra. We discuss various examples; in particular, we deduce that the category of chain complexes with chain equivalences and the category of categories with equivalences are symmetric monoidal proper closed model categories. Eine überarbeitete Fassung ging am 5. 12. 2001 ein     

A Cartan-Eilenberg approach to homotopical algebra

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence of categories between its localisation with respect to weak equivalences and the relative localisation of the subcategory of cofibrant objects with respect to strong equivalences. This equivalence of categories allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.     

Convex hyperspaces of probability measures and extensors in the asymptotic category

The objects of the Dranishnikov asymptotic category are proper metric spaces and the morphisms are asymptotically Lipschitz maps. In this paper we provide an example of an asymptotically zero-dimensional space (in the sense of Gromov) whose space of compact convex subsets of probability measures is not an absolute extensor in the asymptotic category in the sense of Dranishnikov.     

Intrinsic characterization of Alexander-Spanier cohomology groups of compactifications

In this paper we construct a uniform Alexander-Spanier cohomology functor from the category of pairs of uniform spaces to the category of abelian groups. We show that this functor satisfies all Eilenberg-Steenrod axioms on the category of pairs of precompact uniform spaces, is precompact uniform shape invariant and intrinsically, in terms of uniform structures, describes the Alexander-Spanier cohomology groups of compactifications of completely regular spaces.     

Obstruction theory in model categories

Many examples of obstruction theory can be formulated as the study of when a lift exists in a commutative square. Typically, one of the maps is a cofibration of some sort and the opposite map is a fibration, and there is a functorial obstruction class that determines whether a lift exists. Working in an arbitrary pointed proper model category, we classify the cofibrations that have such an obstruction theory with respect to all fibrations. Up to weak equivalence, retract, and cobase change, they are the cofibrations with weakly contractible target. Equivalently, they are the retracts of principal cofibrations. Without properness, the same classification holds for cofibrations with cofibrant source. Our results dualize to give a classification of fibrations that have an obstruction theory.     

Models for homotopy n-types in diagram categories

The classical Mac Lane-Whitehead equivalence showing that crossed modules of groups are algebraic models of connected homotopy 2-types has found a corresponding equivariant version by Moerdijk and Svensson ([22]). In this paper we show that this equivariant result has a higher-dimensional version which gives an equivalence between the homotopy category of diagrams of certain objects indexed by the orbit category of a group H and H-equivariant homotopy n-types for n1.Supported by DGICYT:PS90-0226