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Noncommutative topological forms. (Formes topologiques non commutatives.) (French) Zbl 0837.55004
From the abstract: “Let \(\mathbb{X}\) be a simplicial complex with a base point \(\ast\) and let \(L (\mathbb{X})\) be the free abelian group, generated by \(\mathbb{X}\) with the relation \(\ast = 0\). As it is well-known (Dold-Thom), the homotopy groups of \(L (\mathbb{X})\) are naturally the reduced homotopy groups of \(\mathbb{X}\). The purpose of this paper is to relate this classical result with the theory of noncommutative differential forms.
The author defines a noncommutative de Rham complex for an arbitrary CW- complex. Its elements are noncommutative differential forms with values in a commutative ring. With such noncommutative differential forms the author develops the formalism of ordinary cohomology theory, including cup-product, Steenrod powers and analogs of Thomas-Pontrjagin powers”.
Reviewer: V.Moroz (Minsk)

MSC:
55N35 Other homology theories in algebraic topology
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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