Novikov and Borel conjectures

Classifying manifolds and spaces similar to manifolds up to homeomorphism is a major goal of topology. For lines and surfaces this classification is available. It can be described very reliably using pictures on paper. A lot of work in three-dimensional and even higher-dimensional topology is also very pictorial. A potentially more powerful approach is by using functors from algebraic topology such as the fundamental group or homology. Unfortunately, most of the known functors cannot distinguish between homotopy equivalent spaces. The general purpose of topological surgery theory is to address this point. Suppose it is known that the two manifolds are homotopy equivalent, what can be done to further distinguish between them? The procedure is very constructive: try to manipulate one manifold into the other via special operations through several stages while watching out for obstructions to doing that. The potential obstructions are themselves organized into groups. Now if the appropriate group can be shown to be trivial, there is no obstruction to surgery. If the group is not trivial, there is still hope since the actual obstructions in the group might be trivial but that kind of treatment usually requires agressive intervention with an experienced surgeon at hand.

The rigidity conjecture is about a class of manifolds which should never require surgery. They are the closed aspherical manifolds, that is, compact manifolds without boundary whose universal cover is contractible. Whether two such manifolds are homotopy equivalent can be detected already by comparing the fundamental groups.

Topological Rigidity a.k.a. Borel Conjecture. If two closed aspherical manifolds are homotopy equivalent then they are, in fact, homeomorphic.

This is true for lines and surfaces—all of them are aspherical except for the sphere and the projective plane. For higher-dimensional manifolds this conjecture is actually stronger than the celebrated Poincare conjecture which states that any closed manifold homotopy equivalent to a sphere must be homeomorphic to the sphere.

Examples. There are many examples of aspherical manifolds such as closed nonpositively curved Riemannian manifolds, nilmanifolds and solvmanifolds, Davis' manifolds not covered by Euclidean spaces, asphericalizations of arbitrary closed manifolds a la Gromov. Many more important examples of aspherical manifolds come with boundary or with corners such as compactified arithmetic quotients of symmetric spaces, and there are relative rigidity conjectures for such manifolds.

There are more refined versions of the Novikov and Borel conjectures stated in R. Kirby's List of Problems in Low-dimensional Topology, in Geometric Topology (W.H. Kazez, ed.), Studies in Adv. Math., Vol.2, Part 2, AMS (1997), pp. 355-358. This refers to the very last problem 5.29, one of several which are "high-dimensional". This 380 page list is available in PostScript from Kirby's web page at Berkeley. Here is the PDF file of the relevant pages 278-281.

For a brief discussion of topology of manifolds which is the context for the conjectures, including more refined statements, see Shmuel Weinberger's review [Bull. Amer. Math. Soc. 33 (1996), pp. 93-99] of the book by Andrew Ranicki, Algebraic L-theory and topological manifolds, Cambridge Tracts in Math., Vol. 102, Cambridge Univ. Press (1992). The review is available on-line from AMS.

Here is the link to the search of the AMS Math Reviews database for Novikov conjecture. The search for Borel conjecture returns an amusing mixture of topology and set theory.

A very useful resource are the two volumes of proceedings on "Novikov conjectures, index theorems and rigidity" including papers from the conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, September 6-10, 1993. Edited by Steven C. Ferry, Andrew Ranicki and Jonathan Rosenberg. London Mathematical Society Lecture Note Series, 226-227. Cambridge University Press, Cambridge, 1995. Links to the contents: Vol. 1 and Vol. 2. Another good survey is by S. Weinberger, Aspects of the Novikov Conjecture, Contemp. Math. 105 (1990), 281-297. The recent book by F.T. Farrell, Surgical methods in rigidity, Tata Institute lecture notes (1998), is very readable and gets to recent results in just 60 pages.

Links? The only other web page I know which is directly related to NC is the Novikov Conjecture Home Page of Jonathan Rosenberg. One might expect the address www.novikov.com to lead to the mystical "primal Novikov site"; disappointingly, the site is in fact the home page of a Moscow based design studio. It is pretty cool in its own way though. And a telltale sign of the coming ubiquity of real-life applications of the two conjectures is www.borel.com. It inappropriately touts "flexibility" betraying the idea behind their success.