The aim of this book is to
introduce hyperbolic geometry and its applications to two- and three-manifolds
topology. Topics covered includes: Hyperbolic geometry, Hyperbolic space,
Hyperbolic manifolds, Thick-thin decomposition, The sphere at infinity,
Surfaces, Teichmuller space, Topology of three-manifolds, Seifert manifolds,
Constructions of three-manifolds, Three-manifolds, Mostow rigidity theorem,
Hyperbolic Dehn filling.

This book
covers the following topics: Cohomology and Euler Characteristics Of Coxeter
Groups, Completions Of Stratified Ends, The Braid Structure Of Mapping Class
Groups, Controlled Topological Equivalence Of Maps in The Theory Of Stratified
Spaces and Approximate Fibrations, The Asymptotic Method In The Novikov
Conjecture, N Exponentially Nash G Manifolds and Vector Bundles, Controlled
Algebra and Topology.

This note covers some topics related to the
classification of manifolds. The emphasis will be on manifolds of low dimension
and cases where it is possible to obtain very precise information.

The book is
divided into two parts, called Algebra and Topology. In principle, it is
possible to start with the Introduction, and go on to the topology in Part II, referring back to Part I for novel algebraic concepts.

The intent of this lecture note is to describe the very strong
connection between geometry and lowdimensional topology in a way which will
be useful and accessible to graduate students and mathematicians working in
related fields, particularly 3-manifolds and Kleinian groups.