Embedding 1-connected manifolds

5-manifolds: 1-connected*

We recall the classification of closed smooth simply-connected 5manifolds begun by Smale in 1962 and completed by Barden in 1965. We also give a construction of every such manifold which differs from Barden’s original construction.

Embedding arithmetic hyperbolic manifolds

We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n+ 1)-manifold or its universal mod 2 abelian cover can.

Embedding Multiple Manifolds

Many high dimensional data sets are characterized by a small number of degrees of freedom. Standard manifold-based dimensionality reduction techniques include the bias that all data lies on (or near) a single global manifold. Here we consider data sampled from multiple manifolds. We demonstrate a straightforward technique for multiple manifold embedding: the data is clustered, and each cluster ...

The Classification of (n− 1)-connected 2n-manifolds

This is the question Milnor was actually working as a long-term project during the 1950’s when he “accidentally” discovered exotic spheres, as he explains in [Mil]. This was before the breakthrough of Smale and the Poincaré Conjecture still seemed completely out of reach. Somehow this was the “next easiest” problem in manifold topology and possibly ammenable to the extraordinary tools already d...

Isometric Embedding of Riemannian Manifolds