Homotopy skein modules of orientable 3-manifolds

We define the homotopy skein module of an arbitrary orientable 3-manifold M. This module is similar to the ordinary skein module defined by the second author but is more appropriate when considering oriented links in M up to link homotopy rather than isotopy. We compute the homotopy skein module of M = F × I for any orientable surface F and show that it is free. In the case where M = F × I the homotopy skein module may be given an algebra structure and we show that as an algebra it is isomorphic to the universal enveloping algebra of the Goldman–Wolpert Lie algebra of F. We show, also in this case, that the homotopy skein module is a quantization of the symmetric tensor algebra associated to the Goldman–Wolpert Lie algebra.