Notes of course at IHP: Applications of noncommutative topology in geometry and string theory
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چکیده
ing from the examples we have seen,topologists define twisted cohomology theoriesas follows. In a reasonable category of spaces(say those homotopy equivalent to CW com-plexes), any cohomology theory X 7→ E∗(X) isrepresentable by a representing object E calleda spectrum. (There is no connection withthe spectrum of an element of a Banach al-gebra, or the word “spectrum” meaning “dualspace” for a C∗-algebra.) So, for example,E0(X) = [X,E], meaning homotopy classes of(based) maps. A twisted E-group of X will beE0E(X) = π0 (Γ(X, E)), where E is a (possiblynon-trivial) “principal E-bundle” over X, i.e.,there is a fibration
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Notes on Noncommutative Geometry
Noncommutative geometry has roots in and is a synthesis of a number of diverse areas of mathematics, including: • Hilbert space and single operator theory; • Operator algebras (C*-algebras and von Neumann algebras); • Spin geometry – Dirac operators – index theory; • Algebraic topology – homological algebra. It has certainly also been inspired by quantum mechanics, and, besides feedback to the ...
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