  Cohen and Godin constructed positive boundary topological quantum field
theory (TQFT) structure on the homology of free loop spaces of oriented closed
smooth manifolds by associating a certain operations called string operations
to orientable surfaces with parametrized boundaries. We show that all TQFT
string operations associated to surfaces of genus at least one vanish
identically. This is a simple consequence of properties of the loop coproduct
which will be discussed in detail. One interesting property is that the loop
coproduct is nontrivial only on the degree $d$ homology group of the connected
component of $LM$ consisting of contractible loops, where $d=\dim M$, with
values in the degree 0 homology group of constant loops. Thus the loop
coproduct behaves in a dramatically simpler way than the loop product.

Algebraic Topology