Topological and Bivariant K-Theory pp 141-172 | Cite as

# A survey of bivariant K-theories

## Abstract

- 1.
Gennadi Kasparov’s KK — constructed from “generalised elliptic operators.” This was the first bivariant K-theory to be developed and works for

*C**-algebras [71]. Kasparov’s theory has been adapted to take into account symmetries such as group actions [73] and groupoid actions [77]. - 2.
BDF-Kasparov Ext — constructed from extensions of

*C**-algebras by a stable*C**-algebra, modulo split extensions. The original BDF (Brown-Douglas-Fillmore) one-variable version of [23] is constructed from*C**-algebra extensions by \( \mathcal{K} \) . - 3.
Algebraic Dual K-Theory — an algebraic analogue of one-variable Ext. This is the easiest of these theories to define.

- 4.
Homotopy-Theoretic KK — an analogue of KK constructed using homotopy theory, with a “built-in UCT.”

- 5.
Connes-Higson E-Theory — A simpler replacement for KK, devised by Alain Connes and Nigel Higson [30], designed to eliminate certain technical difficulties that arise when working with non-nuclear

*C**-algebras. This often agrees with Kasparov’s theory and is somewhat easier to define; this theory also admits equivariant versions for groups and groupoids.

## Keywords

Exact Sequence Universal Property Inductive Limit Mapping Cone Hilbert Module## Preview

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