Our aim is to show a method of finding all natural transformations of a functor $T{T}^{*}$ into itself. We use here the terminology introduced in [4,5]. The notion of a soldered double linear morphism of soldered double vector spaces (fibrations) is defined. Differentiable maps $f:C_0\to C_0$ commuting with $T{T}^{*}$-soldered automorphisms of a double vector space $C_0=V^{*}\times V\times V^{*}$ are investigated. On the set $Z_s\left(C_0\right)$ of such mappings, appropriate partial operations are introduced. The natural transformations $T{T}^{*}\to T{T}^{*}$ are bijectively related with the elements...

Let E be the total space of a Hurewicz fiber space whose base and all fibers are ANRs. We prove that if E is metrisable, then it is also an ANR.

The fourth axiom of a model category states that given a commutative square of maps, say i: A → B, g: B → Y, f: A → X, and p: X → Y such that gi = pf, if i is a cofibration, p a fibration and either i or p is a weak equivalence, then a lifting (i.e. a map h: B → X such that ph = g and hi = f) exists. We show that for many model categories the two conditions that either i or p above is a weak equivalence can be embedded in an infinite number of conditions which imply the existence of a lifting (roughly,...

Nous construisons un feuilletage exotique de classe $C^1$ sur tout fibré hyperbolique de genre $1$. Nous montrons égalemnt des théorèmes de rigidité des feuilletages modèles sur certains fibrés pseudo-Anosov.

M. Steinberger et J. West ont prouvé dans [7] qu’un fibré de Serre p:E → B entre CW-complexes a la propriété de relèvement des homotopies par rapport aux k-espaces. Malheureusement, leur démonstration contient une légère erreur. Ils affirment que certains ensembles (notés U et $p^{-1}U\times U$) sont des CW-complexes car ce sont des ouverts de CW-complexes. Ceci est généralement faux, et notre premier objectif dans cette note est de donner des exemples d’ouverts de CW-complexes n’admettant aucune décomposition...