Cyclic vs mixed homology
Ulrich Kraehmer, Dylan Madden
TL;DR
This paper extends the spectral theory of the Karoubi operator to mixed complexes, analyzing how perturbations by Connes' coboundary map affect homology through exact sequences.
Contribution
It introduces a generalized spectral framework for mixed complexes and examines the homological effects of perturbations by Connes' map.
Findings
Extension of Karoubi operator spectral theory to mixed complexes
Homological impact characterized by short exact sequences
Perturbation analysis of Connes' coboundary map
Abstract
The spectral theory of the Karoubi operator due to Cuntz and Quillen is extended to general mixed (duchain) complexes, that is, chain complexes which are simultaneously cochain complexes. Connes' coboundary map B can be viewed as a perturbation of the noncommutative De Rham differential d by a polynomial in the Karoubi operator. The homological impact of such perturbations is expressed in terms of two short exact sequences.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Operator Algebra Research