| Summary:
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This paper gives an exposition of algebraic K-theory, which studies functors $K_n:\text{Rings}\to \text{Abelian Groups}$, $n$ an integer. Classically $n=0,1$ introduced by Bass in the mid 60's (based on ideas of Grothendieck and others) and $n=2$ introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite, Whitehead torsion which classifies $h$-cobordism for closed manifolds of dimension $\geq 5$, and the Hatcher-Wagoner theorem on pseudo-isotopy of differentiable manifolds) are briefly described. Furthermore it is explained in terms of exact sequences and products how the functors $K_i$ are connected. In the mid 1970's Quillen, using methods of homotopy theory, introduced functors $K_n$ for $n$ an arbitrary non-neg! () |
