Non-commutative Topology

Abstract

We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex \((\tilde {b}, d)\); the boundary \(\tilde {b}\) is called topological Hochschild boundary. These ideas combine Connes’ work (Noncommutative geometry, Academic Press, 1994) with ideas of the articles by Teleman and Teleman (St Cerc Math Tom 18(5):753–762, Bucarest, 1966; Rev Roumaine Math Pures Applic 12:725–731, 1967; Central European journal of mathematics. C*-algebras and elliptic theory II. Trends in mathematics, 251–265, Birkhauser, 2008; Local3 index theorem. arXiv: 1109.6095v1, math.KT, 28 Sep. 2011; textitLocal Hochschild homology of Hilbert-Schmidt operators on simplicial spaces. arXiv hal-00707040, Version 1, 11 June 2012; Local algebraic K-theory, 26 Aug 2013 - arXiv.org > math > arXiv:1307.7014, 2013; The local index theorem, HAL-00825083, arXiv 1305.5329, 22 May 2013; \(K_ {i}^{loc}(\ mathbb {C}) \), i = 0, 1 10 set 2013 - arXiv:1309.2421v1 [math.KT] 10 Sep 2013).