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The algebraic K-theory of M is a space A(M) which is defined so that it plays essentially the same role for higher K-groups as K 1 (Zπ 1 (M)) does for M. In particular, Waldhausen showed that there is a map from A ( M ) to a space Wh( M ) which generalizes the map K 1 ( Z π 1 ( M )) → Wh( π 1 ( M )) and whose homotopy fiber is a homology ...
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme.In algebraic topology, it is a cohomology theory known as topological K-theory.
In the mid-seventies, he extended the connection between geometric topology and algebraic K-theory by introducing A-theory, a kind of algebraic K-theory for topological spaces. This led to new foundations for algebraic K-theory (using what are now called Waldhausen categories) and also gave new impetus to the study of highly structured ring ...
In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms. If C has a structure of an exact category, then by defining we(C) to be isomorphisms, co(C) to be admissible monomorphisms, one obtains a structure of a Waldhausen category on C.
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.
The K-groups of finite fields are one of the few cases where the K-theory is known completely: [2] for , = (() +) {/ (), =,For n=2, this can be seen from Matsumoto's theorem, in higher degrees it was computed by Quillen in conjunction with his work on the Adams conjecture.
Assembly maps are studied in geometric topology mainly for the two functors (), algebraic L-theory of , and (), algebraic K-theory of spaces of . In fact, the homotopy fibers of both assembly maps have a direct geometric interpretation when X {\displaystyle X} is a compact topological manifold.
In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory.More precisely, given an exact category C, the construction creates a topological space + so that (+) is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for =,,, (+) is the i-th K-group of R in the classical sense.