Quillen's work in algebraicK-theory

Abstract

Abstract. We survey the genesis and development of higher algebraic K-theory by Daniel Quillen.

Global algebraic K‐theory

“…After some attempts to extend the Grothendieck group to higher K‐groups by purely algebraic means, several people proposed definitions of higher K‐groups as homotopy groups of certain K‐theory spaces at the end of the 60's; in retrospect, the most successful definition is due to Quillen, originally introduced via the so‐called ‘plus construction’ [31]. Several excellent survey papers on the early days of K‐theory are available [11, 44], and we refer to these references for more details and proper credit. Higher algebraic K‐groups are powerful invariants (albeit notoriously difficult to compute) that contain arithmetic information about rings, and geometric information about symmetries of high‐dimensional manifolds.…”

Global algebraic K-theory

“…After some attempts to extend the Grothendieck group to higher K-groups by purely algebraic means, several people proposed definitions of higher K-groups as homotopy groups of certain K-theory spaces at the end of the 60's; in retrospect, the most successful definition is due to Quillen, originally introduced via the so called 'plus construction' [29]. Several excellent survey papers on the early days of K-theory are available [11,42], and we refer to these references for more details and proper credit. Higher algebraic K-groups are powerful invariants (albeit notoriously difficult to compute) that contain arithmetic information about rings, and geometric information about symmetries of high-dimensional manifolds.…”