Novikov Conjectures, Index Theorems and Rigidity
Oberwolfach 1993
Volume 1
Edited by Steven C. Ferry
Edited by Andrew Ranicki
Edited by Jonathan M. Rosenberg
Publisher: Cambridge University Press
Print Publication Year: 1995
Online Publication Date:May 2010
Online ISBN:9780511662676
Paperback ISBN:9780521497961
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662676.003
Subjects: Geometry and Topology
Image View Extract Fullview: Text View | Enlarge Image ‹ Previous Chapter ›Next Chapter
Precursors of the Novikov Conjecture
Characteristic classes
The Novikov Conjecture has to do with the question of the relationship of the characteristic classes of manifolds to the underlying bordism and homotopy theory. For smooth manifolds, the characteristic classes are by definition the characteristic classes of the tangent (or normal) bundle, so basic to this question is another more fundamental one: how much of a vector bundle is determined by its underlying spherical fibration? The Stiefel-Whitney classes of vector bundles are invariants of the underlying spherical fibration, and so the Stiefel-Whitney numbers of manifolds are homotopy invariants. Furthermore, they determine unoriented bordism. The Pontrjagin classes of vector bundles are not invariants of the underlying spherical fibration, and the Pontrjagin numbers of manifolds are not homotopy invariants. However, together with the Stiefel-Whitney numbers, they do determine oriented bordism. The essential connection between characteristic numbers and bordism was established by Thom [Th1] in the early 1950's.
Geometric rigidity
As we shall see later, the Novikov Conjecture is also closely linked to problems about rigidity of aspherical manifolds. As everyone learns in a first course in geometric topology, closed 2-manifolds are determined up to homeomorphism by their fundamental groups. In higher dimensions, of course, nothing like this is true in general, but one can still ask if aspherical closed manifolds (closed manifolds having contractible universal cover) are determined up to homeomorphism by their fundamental groups.
pp. i-vi
pp. vii-viii
pp. ix-x
Introductory Material : Read PDF
pp.
Programme of the Conference : Read PDF
pp. 1-2
Participant List from the Conference : Read PDF
pp. 3-6
A history and survey of the Novikov conjecture : Read PDF
pp. 7-66
Annotated Problem List : Read PDF
pp. 67-78
pp.
On the Steenrod homology theory (first distributed 1961) : Read PDF
pp. 79-96
Homotopy type of differentiable manifolds (first distributed 1962) : Read PDF
pp. 97-100
K-theory, group C*-algebras, and higher signatures (Conspectus) (first distributed 1981) : Read PDF
pp. 101-146
Research and Survey Papers : Read PDF
pp.
A coarse approach to the Novikov conjecture : Read PDF
pp. 147-163
Geometric reflections on the Novikov conjecture : Read PDF
pp. 164-173
Controlled Fredholm representations : Read PDF
pp. 174-200
Assembly maps in bordism-type theories : Read PDF
pp. 201-271
On the Novikov conjecture : Read PDF
pp. 272-337
Analytic Novikov for topologists : Read PDF
pp. 338-372