Stable Equivalence of Differentiable Manifolds

Some problems in differential geometry and topology

The general setting here is that one would like to understand the classification of manifolds, and related objects such as knots in 3-space. The “classification”means up to an appropriate equivalence, and for definiteness we will consider smooth (i.e. differentiable) manifolds up to diffeomorphism (equivalence by differentiable homeomorphisms). Of course, what one really seeks is an understandi...

The Differentiable Chain Functor Is Not Homotopy Equivalent to the Continuous Chain Functor

Let S∗ and S ∞ ∗ be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation i : S∞ ∗ −→ S∗, which induces homology equivalences over each manifold, is not a natural homotopy equivalence.

Stable Exponentially Harmonic Maps Between Finsler Manifolds

A Survey of Some Recent Developments in Differential Topology

1. We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse. For convenience, differentiable means C; in the problems we consider, C' would serve as well. The notions of different...

Differentiable structures on the 7-sphere

The problem of classifying manifolds has always been central in geometry, and different categories of manifolds can a priori give different equivalence classes. The most famous classification problem of all times has probably been the Poincare’ conjecture, which asserts that every manifold which is homotopy equivalent to a sphere is actually homeomorphic to it. A large family of invariants have...