Brown representability for exterior cohomology and cohomology with compact supports

Abstract

It is well known that cohomology with compact supports is not a homotopy invariant but only a proper homotopy one. However, as the proper category lacks of general categorical properties, a Brown representability theorem type does not seem reachable. However, by proving such a theorem for the so called exterior cohomology in the complete and cocomplete exterior category, we show that the n-th cohomology with compact supports of a given countable, locally finite, finite dimensional relative CW-complex (X, R + ) is naturally identified with the set [X, K n ] R+ of exterior based homotopy classes from a "classifying space" K n . We also show that this space has the exterior homotopy type of the exterior Eilenberg-MacLane space for Brown-Grossman homotopy groups of type (R ∞ , n), R being the fixed coefficient ring.

Classification of exterior and proper fibrations

“…Finally, a space X ∈ E R + is exterior path connected if it is path connected as a topological space and [S 0 + , X] R + = { * }. Here, S 0 + denotes R + with a 0-sphere attached to each integer number, endowed with the cocompact 1 We warn the reader that this category was denoted by E R + w in [9,10] while E R + was reserved for based exterior spaces, non necessarily well pointed, i.e., the based ray is not necessarily a closed cofibraton. As all based exterior spaces we consider here are well pointed we avoid excessive notation.…”