Modules associated to disconnected surfaces by quantization functors

Blanchet, Habegger, Masbaum and Vogel defined a quantization functor on a category whose objects are oriented closed surfaces together with a collection of colored banded points and a p1-structure. The functor assigns a module Vp(Σ) to each surface Σ. This assignment satisfies certain axioms. For p even, it satisfies the tensor product axiom, which gives the modules associated to a disconnected surface as the tensor product of the modules associated to its components. We show that the p odd case satisfies a generalized tensor product formula. The notion of a generalized tensor product formula is due to Blanchet and Masbaum. We let V̂p(Σ) denote Vp(Σ⨿Ŝ2), where Ŝ2 is a sphere with one banded point colored p−2. The generalized tensor product formula expresses Vp(Σ1⨿Σ2) in terms of Vp(Σ1), Vp(Σ2), V̂p(Σ1) and V̂p(Σ2). We reduce the calculation of V̂p(Σ) to known results, and calculate V̂p(Σ) explicitly in many cases.