Product structures on the mapping space from the torus to the 3-dimensional sphere

product structures on topological spaces. For a given product structure
the natural questions arise wether this product is associative or
commutative. In the case of the mapping space from the torus to the
3-dimensional sphere it can be shown that the mapping space splits into
a product of simpler spaces, called loop spaces. It turns out that the
question of commutativity of the mapping space is closely connected to
certain maps between the loop spaces. This might help to get a better
insight in the product structure of the above mapping space.