In the 1950s, Eilenberg and Steenrod presented their
famous characterization of homology theory by seven axioms. Somewhat later, it
was found that keeping just the first six of these axioms (all except the
condition on the “homology” of the point), one can obtain many
other interesting systems of algebraic invariants of topological manifolds,
such as $K$-theory, cobordisms, and others. These theories come under
the common name of generalized homology (or cohomology) theories.
The purpose of the book is to give an exposition of generalized (co)homology
theories that can be read by a wide group of mathematicians who are not experts
in algebraic topology. It starts with basic notions of homotopy theory and then
introduces the axioms of generalized (co)homology theory. Then the authors
discuss various types of generalized cohomology theories, such as
complex-oriented cohomology theories and Chern classes, $K$-theory,
complex cobordisms, and formal group laws. A separate chapter is devoted to
spectral sequences and their use in generalized cohomology theories.
The book is intended to serve as an introduction to the subject for
mathematicians who do not have advanced knowledge of algebraic topology.
Prerequisites include standard graduate courses in algebra and topology, with
some knowledge of ordinary homology theory and homotopy theory.
Readership
Graduate students and research mathematicians interested in
algebraic topology.
