We establish Whitehead and Ganea characterizations for proper LS-category. We use the embedding of the proper category into the exterior category, and construct in the latter a suitable closed model structure of Strøm type. Then, from the axiomatic LS-category arising from the exterior homotopy category we can recover the corresponding proper LS invariants. Some applications are given.

We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.

Link to this article: http://journals.cambridge.org/abstract_S0305004110000174 How to cite this article: JOSE M. GARCÍA-CALCINES, PEDRO R. GARCÍA-DÍAZ and ANICETO MURILLO MAS (2010). The Ganea conjecture in proper homotopy via exterior homotopy theory. Mathematical AbstractIn this article we provide sufficient conditions on a space X to verify Ganea conjecture with respect to exterior and proper Lusternik-Schnirelmann category. For this aim we previously develop an exterior version of the Whitehead, cellular approximation, CWapproximation and Blakers-Massey theorems within a homotopy theory of exterior CWcomplexes and study their corresponding analogues and consequences in the proper setting.

An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex S can be realized as the k-skeleton of some elliptic complex as long as k > dim S. A functorial version of this conjecture due to McGibbon is that for any n there exists an elliptic complex E n and an n-equivalence S → E n . In fact, this is equivalent to its Eckmann-Hilton dual, which we prove in the rational category for a small class of simply connected spaces. Moreover, we construct the n-equivalence in such a way that the homotopy fibre is, rationally, a product of a finite number of odd spheres.

An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex $S$ can be realized as the $k$-skeleton of some elliptic complex as long as $k\,{>}\,\dim S$, or, equivalently, that any simply connected finite Postnikov piece $S$ can be realized as the base of a fibration $F\,{\to}\,E\,{\to}\,S$ where $E$ is elliptic and $F$ is $k$-connected, as long as the $k$ is larger than the dimension of any homotopy class of $S$. This conjecture is only known in a few cases, and here we show that in particular if the Postnikov invariants of $S$ are decomposable, then the Anick conjecture holds for $S$. We also relate this conjecture with other finiteness properties of rational spaces.