Non-embedding theorems for Y-spaces

Abstract

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).