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Foundations of geometric cohomology: from co-orientations to product structures (2212.07482v2)

Published 14 Dec 2022 in math.AT and math.GT

Abstract: This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product of such smooth maps, which provides our geometric cochains with a partially defined product structure inducing the cup product in cohomology. A parallel treatment of homology is also given allowing for a geometric unification of the contravariant and covariant theories.

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