Using h-Cobordisms to Detect Non-Trivial Homotopy Groups in Spaces of Legendrian

In this talk I will first define the space of h-cobordisms associated to a manifold M. This space is known to have many non-trivial homotopy groups and in stable range (they can often be computed using Waldhausen's algebraic K-theory of spaces). I will then define maps from these spaces into certain spaces of Legendrians, and I will describe how we detect that these maps are to some extend non-trivial on homotopy groups. A special case is that we essentially (and in stable range) find two copies of the h-cobordism space of a point inside the space of Legendrians of the jet 1 bundle of the n-sphere based at a standard Legendrian unknot (the Whitney sphere).