Pseudoholomorphic punctured spheres in R×(S1×S2): Properties and existence

This is the first of at least two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in R× (S1 × S2) as defined by a certain natural pair of almost complex structure and symplectic form. This article proves that all moduli space components are smooth manifolds. Necessary and sufficient conditions are also given for a collection of closed curves in S1 × S2 to appear as the set of |s| → ∞ limits of the constant s ∈ R slices of a pseudoholomorphic, multiply punctured sphere.