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Cup i-product on the communitative differential forms and Steenrod squares. (Cup i-produit sur les formes différentielles non commutatives et carrés de Steenrod.) (English) Zbl 1117.55015

The author defines cup-\(i\) products in the context of non-commutative differential forms. Exactly analogous to N. E. Steenrod’s original cup-\(i\) products in cochain complexes [Ann. Math. 48, 290–320 (1947; Zbl 0030.41602)], these are maps \(\Omega^p(A) \otimes \Omega^q(A) \to \Omega^{p+q-i}(A)\) where \(A\) is a unitary algebra and \(\Omega^n(A)\) is the set of non-commutative differential forms of degree \(n\). These cup-\(i\) products provide a very explicit construction of Steenrod operations in this setting, complementing the axiomatic approach already established by M. Karoubi [Topology 34, 699–715 (1995; Zbl 0834.55011)]. They also provide a generalization of the \(b\) operator of Hochschild and cyclic homology.

MSC:

55S10 Steenrod algebra
58A10 Differential forms in global analysis
18G55 Nonabelian homotopical algebra (MSC2010)
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