Manifolds: Topological versus Differentiable∗
Here are some facts about the differential (smooth) structures that a topological manifold
of dimension ncan support, and related results.
Unless stated otherwise, all manifolds are without boundary. Counting (e.g., uniqueness)
is meant in the appropriate sense (up to smooth diffeomorphisms for smooth manifolds).
See also the remarks on pages 14 and 37 of the textbook.
(a) n≤3
•Each n-dimensional topological manifold has a unique smooth structure
(J. Munkres, E. Moise).
(b) n≥4
•For each nthere is a connected compact topological manifold that does not admit
a smooth structure.
•Each n-dimensional compact connected topological manifold admits at most
countably many different smooth structures.
•S7admits exactly 28 different smooth structures (J.W. Milnor and M.A. Ker-
vaire).
•Classification of smooth (even compact) manifolds of dimension n≥4 is very
hard.
•Does S4or CP2admit more than one smooth structures?
(c) The Poincar´e conjecture (1904): a compact simply-connected1smooth manifold is
homeomorphic to the sphere of that dimension.
•n≥5 proved by S. Smale (1961).
•n= 4 proved by M.H. Freedman (1982).
•n= 3 proved by G. Perelman (2002–2003), using the Ricci flow method of
R.S. Hamilton.
This makes the “geometrization conjecture” of W. Thurston close to being
proved2: there are eight standard Riemannian models in dimension 3 (compare
to the dimension-2 case below, where there are three models).
∗Last updated: Jan 27, 2009; only limited claim of accuracy is made.
1A (path connected) topological space Xis simply-connected if its fundamental group, π1(X), is trivial:
any closed path in Xcan be continuously “shrunk” to a point. The 1-sphere is not simply-connected but all
higher dimensional spheres are.
2Might be already proven, using Perelman’s work.
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