Cup-product e prodotto esterno
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Alternating cup-product for alternating singular cochains and cohomology classes, over a field of characteristic zero, is considered. Differently from ordinary cup-product, the alternating one is skew-commutative for cochains also, and exactly parallel to exterior product for differential forms. A direct proof of de Rham's third theorem follows, with use of a convenient simplicial (but not barycentric) subdivision.