|
Showing 1 - 3 of
3 matches in All Departments
Some Historical Background This book deals with the cohomology of
groups, particularly finite ones. Historically, the subject has
been one of significant interaction between algebra and topology
and has directly led to the creation of such important areas of
mathematics as homo logical algebra and algebraic K-theory. It
arose primarily in the 1920's and 1930's independently in number
theory and topology. In topology the main focus was on the work
ofH. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among
others) made significant contributions. The main thrust of the
early work here was to try to understand the meanings of the low
dimensional homology groups of a space X. For example, if the
universal cover of X was three connected, it was known that H2(X;
A. ) depends only on the fundamental group of X. Group cohomology
initially appeared to explain this dependence. In number theory,
group cohomology arose as a natural device for describing the main
theorems of class field theory and, in particular, for describing
and analyzing the Brauer group of a field. It also arose naturally
in the study of group extensions, N"
An introduction to the theory of orbifolds from a modern
perspective, combining techniques from geometry, algebraic topology
and algebraic geometry. One of the main motivations, and a major
source of examples, is string theory, where orbifolds play an
important role. The subject is first developed following the
classical description analogous to manifold theory, after which the
book branches out to include the useful description of orbifolds
provided by groupoids, as well as many examples in the context of
algebraic geometry. Classical invariants such as de Rham cohomology
and bundle theory are developed, a careful study of orbifold
morphisms is provided, and the topic of orbifold K-theory is
covered. The heart of this book, however, is a detailed description
of the Chen-Ruan cohomology, which introduces a new product for
orbifolds and has had significant impact in recent years. The final
chapter includes explicit computations for a number of interesting
examples.
Some Historical Background This book deals with the cohomology of
groups, particularly finite ones. Historically, the subject has
been one of significant interaction between algebra and topology
and has directly led to the creation of such important areas of
mathematics as homo logical algebra and algebraic K-theory. It
arose primarily in the 1920's and 1930's independently in number
theory and topology. In topology the main focus was on the work
ofH. Hopf, but B. Eckmann, S. Eilenberg, and S. MacLane (among
others) made significant contributions. The main thrust of the
early work here was to try to understand the meanings of the low
dimensional homology groups of a space X. For example, if the
universal cover of X was three connected, it was known that H2(X;
A. ) depends only on the fundamental group of X. Group cohomology
initially appeared to explain this dependence. In number theory,
group cohomology arose as a natural device for describing the main
theorems of class field theory and, in particular, for describing
and analyzing the Brauer group of a field. It also arose naturally
in the study of group extensions, N"
|
|