Cylindrical contact homology as a well-defined homology?

In this talk I will explain how the heuristic arguments sketched in literature since 1999 fail to define a homology theory. These issues will be made clear with concrete examples and we will explore what stronger conditions are necessary to develop a theory without the use of virtual chains or polyfolds in 3 dimensions. It turns out that this can be accomplished by placing strong conditions on the growth rates of the indices of Reeb orbits. In addition we sketch a new approach allowing us to compute cylindrical contact for a large class of examples which admit contact forms that are admissible under the stronger conditions required. This approach is applicable to prequantization spaces and the links of simple singularities.