The Novikov Conjecture: Geometry and Algebra (Oberwolfach Seminars)

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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.

The Novikov Conjecture: Geometry and Algebra (Oberwolfach Seminars) Review

In its simplest form, the Novikov conjecture asserts that if there is a map f 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 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 a stable vector bundle. The discussion of spectra involves the important notion of a `homotopy pushout', which are defined so as to commute with the suspension with the unit circle, and the `homotopy pullback', which commutes with the loop functor. Both homotopy pushouts and pullbacks are homotopy equivalences.

Given a discrete group G, a family of subgroups of G, and an equivariant homology theory with respect to G, after constructing the classifying space of the family of subgroups, the authors want to show that the assembly map induced from the projection of the family to the one-point space is an isomorphism. To understand for which groups this is true, the authors must first define the notion of a G-homology theory. They use the notion of a G-CW-complex that they defined when discussing classifying spaces of families of subgroups to define this homology theory. For a group G and an associative commutative ring Q with unit it consists of a collection of covariant functors from the category of G-CW-pairs to the category of Q-modules indexed by the integers that satisfies the usual properties such as G-homotopy invariance, the existence of a long exact sequences for pairs, and excision. An equivariant homology theory is then a G-homology theory that has a `induction structure', the latter of which is a collection of isomorphisms between the nth G-homology groups and nth homology groups of a group that has a homomorphism into G. The authors then show how to obtain an equivariant homology from a spectrum. Central to their construction is the `orbit category Or(G)' of a group G. The objects of this category are homogeneous G-spaces and the morphisms are G-maps. For a small category C they define a `C-space' to be a functor from C to the set of compactly generated spaces. A `C-spectrum' is a functor from C to the category of spectra. After defining a notion of smash product for a C-space and a C-spectrum, the authors then quote a lemma that illustrates how one can obtain a G-homology theory from an Or(G)-spectrum. In order to obtain an induction structure, the Or(G)-spectrum must be obtained from a spectrum of groupoids. The authors show how to do this and thus obtain an equivariant homology theory. Therefore the K- and L-theory spectra over groupoids that were constructed earlier thus give rise to equivariant homology theories.

These G-homology theories reduce to the K- and L-theory of the group ring when evaluated on a one-point space, and the topological K-theory of the reduced C*-algebra. The Farell-Jones conjecture claims that the assembly maps from the equivariant homology groups for K and L-theory to the one-point homology are isomorphisms. The Baum-Connes conjecture does the same for topological K-theory. If these conjectures were answered positively then they would allow the computation of the K- and L-groups from the K- and L- finite or virtually cyclic subgroups of G. The authors spend two chapters discussing for what groups these conjectures have been found to be true, and also a chapter on how the Novikov conjecture follows from these conjectures.

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