Detecting cohomology classes for the proper LS category. The case of semistable 3-manifolds

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0305004111000417 How to cite this article: MANUEL CÁRDENAS, FRANCISCO F. LASHERAS and ANTONIO QUINTERO (2012). Detecting cohomology classes for the proper LS category. The case of semistable 3-manifolds. Mathematical AbstractWe give sufficient conditions for the existence of detecting elements for the LusternikSchnirelmann category in proper homotopy. As an application we determine the proper LS category of some semistable one-ended open 3-manifolds.

Brown representability for exterior cohomology and cohomology with compact supports

“…We can also consider the exterior cohomology in E, an extension of compactly supported cohomology to the exterior category. This already appears in [25, §2] under a different approach, and it has recently been used to obtain interesting results in proper homotopy theory [7,8,18]. Finally, we show that the exterior cohomology satisfies all the necessary properties to be classified in the Brown sense.…”

“…(ii) For the special case of singular cohomology, exterior cohomology was already introduced in [7] by considering, for each E ∈ E, the complex C * (X, E) of singular cochains of X which vanish in E.…”

“…In this section, as stated in the introduction, we set the main properties of exterior cohomology, already introduced in [25, §2] under a different approach, and successfully used in [7,8,18]. From now on, and unless explicitly stated otherwise, we fix a continuous and additive classical cohomology theory, i.e., a contravariant functor H * from the category of pairs of spaces to the category of non negatively graded abelian groups satisfying exactness, excision, homotopy invariance, continuity (tautness), and additivity.…”

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

“…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.…”