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Birkhäuser - Birkhäuser Mathematics | The Novikov Conjecture - Geometry and Algebra

The Novikov Conjecture

Geometry and Algebra

Series: Oberwolfach Seminars, Vol. 33

Kreck, Matthias, Lück, Wolfgang

2005, XV, 266 p.

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Manifolds are the central geometric objects in modern mathematics. An attempt to understand the nature of manifolds leads to many interesting questions. One of the most obvious questions is the following. Let M and N be manifolds: how can we decide whether M and N are ho- topy equivalent or homeomorphic or di?eomorphic (if the manifolds are smooth)? The prototype of a beautiful answer is given by the Poincar´ e Conjecture. If n N is S ,the n-dimensional sphere, and M is an arbitrary closed manifold, then n it is easy to decide whether M is homotopy equivalent to S . Thisisthecaseif and only if M is simply connected (assumingn> 1, the case n = 1 is trivial since 1 every closed connected 1-dimensional manifold is di?eomorphic toS ) and has the n homology of S . The Poincar´eConjecture states that this is also su?cient for the n existenceof ahomeomorphism fromM toS . For n = 2this followsfromthewe- known classi?cation of surfaces. Forn> 4 this was proved by Smale and Newman in the 1960s, Freedman solved the case in n = 4 in 1982 and recently Perelman announced a proof for n = 3, but this proof has still to be checked thoroughly by the experts. In the smooth category it is not true that manifolds homotopy n equivalent to S are di?eomorphic. The ?rst examples were published by Milnor in 1956 and together with Kervaire he analyzed the situation systematically in the 1960s.

Content Level » Research

Keywords » Algebraische Topologie - Characteristic class - Differentialtopologie - Invariantentheorie - homology - manifold - vector bundle

Related subjects » Birkhäuser Mathematics

Table of contents 

A Motivating Problem.- to the Novikov and the Borel Conjecture.- Normal Bordism Groups.- The Signature.- The Signature Theorem and the Novikov Conjecture.- The Projective Class Group and the Whitehead Group.- Whitehead Torsion.- The Statement and Consequences of the s-Cobordism Theorem.- Sketch of the Proof of the s-Cobordism Theorem.- From the Novikov Conjecture to Surgery.- Surgery Below the Middle Dimension I: An Example.- Surgery Below the Middle Dimension II: Systematically.- Surgery in the Middle Dimension I.- Surgery in the Middle Dimension II.- Surgery in the Middle Dimension III.- An Assembly Map.- The Novikov Conjecture for ?n.- Poincaré Duality and Algebraic L-Groups.- Spectra.- Classifying Spaces of Families.- Equivariant Homology Theories and the Meta-Conjecture.- The Farrell-Jones Conjecture.- The Baum-Connes Conjecture.- Relating the Novikov, the Farrell-Jones and the Baum-Connes Conjectures.- Miscellaneous.- Exercises.- Hints to the Solutions of the Exercises.

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