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Brown representability for exterior cohomology and cohomology with compact supports
Author(s) -
GarcíaCalcines J. M.,
GarcíaDíaz P. R.,
Murillo A.
Publication year - 2014
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdu024
Subject(s) - cohomology , mathematics , de rham cohomology , pure mathematics , equivariant cohomology , čech cohomology , algebra over a field
It is well known that cohomology with compact supports is a proper homotopy invariant. However, as the proper category lacks general categorical properties, a Brown representability theorem type does not seem reachable. 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 based exterior 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.