For other topics of interest to SUNY@Albany faculty, see this page.
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 threedimensional and even higherdimensional 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 higherdimensional 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 Lowdimensional Topology,
in Geometric Topology (W.H. Kazez, ed.), Studies in Adv. Math., Vol.2,
Part 2, AMS (1997), pp. 355358.
This refers to the very last problem 5.29, one of several which are
"highdimensional".
This 380 page list is available in PostScript from
Kirby's web page at Berkeley.
Here is the PDF file of the relevant pages 278281.
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. 9399]
of the book by Andrew Ranicki, Algebraic Ltheory and topological manifolds,
Cambridge Tracts in Math., Vol. 102, Cambridge Univ. Press (1992).
The review is available
online 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 610, 1993.
Edited by Steven C. Ferry, Andrew Ranicki and Jonathan Rosenberg.
London Mathematical Society Lecture Note Series, 226227.
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), 281297.
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 reallife applications of the two conjectures is www.borel.com. It inappropriately touts "flexibility" betraying the idea behind their success.
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bgoldfarb at albany.edu
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home page of Boris Goldfarb
