A revised augmented Cuntz semigroup

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

Countable approximation of topological G-manifolds, II: linear Lie groups G

Let [Formula: see text] be a matrix group. Topological [Formula: see text]-manifolds with Palais-proper action have the [Formula: see text]-homotopy type of countable [Formula: see text]-CW complexes (3.2). This generalizes Elfving’s dissertation theorem for locally linear [Formula: see text]-manifolds (1996). Also, we improve the Bredon–Floyd theorem from compact Lie groups [Formula: see text] to arbitrary Lie groups [Formula: see text].

Homotopy Groups and CW Complexes

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.