Models for Proper homotopy types, (joint work with a research team led by Prof. L. J. Hernandez, Universidad de La Rioja, Spain).

When handling non-compact spaces, ordinary homotopy theory does not reflect enough of the geometric structure of the spaces involved. To replace it, one uses proper homotopy theory. Results from the mid 1970s allow one to embed this theory in the theory of prohomotopy (also used in Strong Shape Theory). This then can use results from homotopy coherence and ``classical'' homotopy theory to obtain new geometric insight into the behaviour of spaces ``at infinity''. New methods of algebraic topology can also be used. The crossed complexes of Brown and Higgins and the Cat^n-groups of Loday allow one to make progress in the direction of a proper form of combinatorial homotopy theory. This has as its aim to provide algebraic models for proper homotopy types so that, at least theoretically, questions on homotopy type, existence or extension of mappings between non-compact complexes can be reduced to algebraic calculations. The methods being developed with Hern\'andez and his collaborators can also be applied to problems in Strong Shape Theory. A substantial survey article on this area has been completed for the ``Handbook of Algebraic Topology ''(ed. I.M.James).

The continuous image of a compact set is compact, so for a non-compact space, it may seem difficult to use the methods of traditional homotopy theory to study it, yet information "at the end" of such a space can be very useful in certain areas of geometric topology. The problem is resolved by using proper maps. These are continuous maps which satisfy an additional condition, namely that the inverse image of a compact subset is compact. Proper homotopy involves the study of invariants of non-compact spaces under proper homotopy equivalence. For instance, by mapping a string of circles into a space by proper maps one obtains a group, similarly if one maps in an infinite cylinder.

"Visualisation of the proper map that links the two main families of properhomotopy groups."

The map that puts the string of circles into the cylinder generates a homomorphism between the corresponding groups which enables one to obtain more information on the groups and also to interpret that information geometrically. The mix of algebraic and geometric intuition that is involved makes this a very exciting area to work in.

(This research has been supported by a grant under the Acciones Integradas scheme.)