A multicurve C on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist tC about C is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism h such that h n = tC. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for n. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4.

We show that the property of being rationally Kstable passes from the fibers of a continuous C(X)-algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers. Given a compact Hausdorff space X, a continuous C(X)-algebra is the section algebra of a continuous field of C*-algebras over X. Such algebras form an important class of non-simple C*-algebras, and it is often of interest to understand those properties of a C*-algebra which pass from the fibers to the ambient C(X)-algebra. Given a unital C*-algebra, we write U n (A) for the group of n × n unitary matrices over A. This is a topological group, and its homotopy groups π j (U n (A)) are termed the nonstable K-theory groups of A. These groups were first systematically studied by Rieffel [20] in the context of noncommutative tori. Thomsen [26] built on this work, and developed the notion of quasi-unitaries, thus constructing a homology theory for (possibly non-unital) C*-algebras. Unfortunately, the nonstable K-theory for a given C*-algebra is notoriously difficult to compute explicitly. Even for the algebra of complex numbers, these groups are naturally related to the homotopy groups of spheres π j (S n), which are not known for many values of j and n. It is here that rational homotopy theory has proved to be useful to topologists and, in this paper, we employ this tool in the context of C*-algebras.

We study the general and connected stable ranks for C*-algebras. We estimate these ranks for pullbacks of C*-algebras, and for tensor products by commutative C*-algebras. Finally, we apply these results to determine these ranks for certain commutative C*-algebras and non-commutative CW-complexes.Stable ranks for C*-algebras were first introduced by Rieffel [16] in his study of the nonstable K-theory of noncommutative tori. A stable rank of a C*-algebra A is a number associated to the C*-algebra, and is meant to generalize the notion of covering dimension for topological spaces. The first such notion introduced by Rieffel, called topological stable rank, has played an important role ever since. In particular, the structure of C*-algebras having topological stable rank one is particularly well understood.Since the foundational work of Rieffel, many other ranks have been introduced for C*algebras, including real rank, decomposition rank, nuclear dimension, etc. In this paper, we return to the original work of Rieffel, and consider two other stable ranks introduced by him: the connected stable rank and general stable rank. The general stable rank determines the stage at which stably free projective modules are forced to be free. The connected stable rank is a related notion, but its definition is less transparent. What links these two ranks, and differentiates them from the topological stable rank, is that they are homotopy invariant.This was highlighted in a paper by Nica [13], where he emphasized the relationship between these two ranks, and how they differ from topological stable rank. Furthermore, in order to compute these ranks for various examples, he showed how they behave with respect to some basic constructions (matrix algebras, quotients, inductive limits, and extensions).

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Given an action of a compact group on a complex vector bundle, there is an induced action of the group on the associated Cuntz--Pimsner algebra. We determine conditions under which this action has finite Rokhlin dimension.