  We introduce a global Cauchy-Riemann($CR$)-invariant and discuss its behavior
on the moduli space of $CR$-structures. We argue that this study is related to
the Smale conjecture in 3-topology and the problem of counting complex
structures. Furthermore, we propose a contact-analogue of Ray-Singer's analytic
torsion. This ``contact torsion'' is expected to be able to distinguish among
``contact lens'' spaces. We also propose the study of a certain kind of
monopole equation associated with a contact structure.
