  Implementing an idea due to John Baez and James Dolan we define new
invariants of Whitney stratified manifolds by considering the homotopy theory
of smooth transversal maps. To each Whitney stratified manifold we assign
transversal homotopy monoids, one for each natural number. The assignment is
functorial for a natural class of maps which we call stratified normal
submersions. When the stratification is trivial the transversal homotopy
monoids are isomorphic to the usual homotopy groups. We compute some simple
examples and explore the elementary properties of these invariants.
  We also assign `higher invariants', the transversal homotopy categories, to
each Whitney stratified manifold. These have a rich structure; they are rigid
monoidal categories for n>1 and ribbon categories for n>2. As an example we
show that the transversal homotopy categories of a sphere, stratified by a
point and its complement, are equivalent to categories of framed tangles.

Algebraic Topology;Category Theory