The Novikov Conjecture: Geometry and Algebra

By Matthias Kreck

These lecture notes contain a guided tour to the Novikov Conjecture and related conjectures due to Baum-Connes, Borel and Farrell-Jones. They begin with basics about higher signatures, Whitehead torsion and the s-Cobordism Theorem. Then an introduction to surgery theory and a version of the assembly map is presented. Using the solution of the Novikov conjecture for special groups some applications to the classification of low dimensional manifolds are given. Finally, the most recent developments concerning these conjectures are surveyed, including a detailed status report.
The prerequisites consist of a solid knowledge of the basics about manifolds, vector bundles, (co-) homology and characteristic classes.

• Format: Paperback
• Language: English
• ISBN: 3764371412
• ISBN13: 9783764371418
• Release Date: November 2004
• Publisher: Birkhauser
• Length: 266 Pages
• Weight: 1.17 lbs.
• Dimensions: 0.8" x 6.8" x 9.4"

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An effective overview

In its simplest form, the Novikov conjecture asserts that if there is a map from a closed smooth manifold M and a classifying space of a group, and if g is a homotopy equivalence from a closed smooth manifold N to M, then the `higher signatures' of (M, f) and (N, fg) agree. The goal of this book is to introduce the reader to the precise notion of `higher signature' and to discuss various concepts and tools used in attempted resolutions of the conjecture. Also discussed in some details are conjectures that are related to the Novikov conjecture. Readers will need expected to have a strong background in algebraic and differential topology in order to appreciate the content of the book, but the authors develop some of the needed material in it, such as h- and s-cobordism, simple homotopy, surgery theory, and the classification problem for manifolds via characteristic classes. Without too many exceptions the authors motivate the concepts exceedingly well, especially in chapter 12 where they give one of the best explanations in print for the surgery obstruction groups. When reading the book it becomes apparent that the Novikov conjecture has many guises, and attempts to resolve it have involved some quite esoteric constructions. The main strategy used in its resolution involves a generalization of the Hirzebruch signature, called a `higher signature' and the notion of an `assembly map.' The assembly map, as the name implies, collects all the higher signatures into a single invariant: essentially the image of the Poincare dual of the L-class under the map induced from f. One then constructs a homomorphism (the assembly map) from the Poincare duals of the Pontrjagin classes to a particular Abelian group L(G), such that the value of the assembly map on the image is a homotopy invariant. The Novikov conjecture is the assertion that the assembly map is an isomorphism. Much of the first part of the book discusses how to make these notions meaningful and how to interpret them geometrically via the surgery obstruction groups. The authors also discuss them in a purely algebraic context, constructing an algebraic notion of bordism in the context of chain complexes and the notions of symmetric and quadratic forms over chain complexes. Algebraic cobordism allows the definition of a symmetric and quadratic algebraic L-group. The nth symmetric algebraic L-group of a ring R with involution is defined as the collection of cobordism classes of n-dimensional symmetric algebraic Poincare complexes, and the quadratic L-group of R, with a similar definition for the quadratic case. From these constructions the reader is introduced to the subject of L-theory, which has been the subject of intense research in the last two decades. Central to the research into the Novikov conjecture is the category of `spectra', which is usually encountered in any treatment of algebraic topology but is discussed here with examples given in K- and L-theory and the famous Thom spectrum of