to closed oriented surfaces with $G$-bundle are nonzero and the vector spaces attached to $G$-bundles over the circle are lines (which are invertible in the collection of vector spaces under tensor product). Now there is an obvious forgetful functor $\pi
{\colon}{\textnormal{Bord}^{SO}_2(G)}\to{\textnormal{Bord}^{SO}}_2$ which omits the $G$-bundle. Then the quantum field theory $F$, obtained by summing over $G$-bundles, can be viewed as a kind of pushforward of $I$ along $\pi$. The relevant pushforward procedure has an analog in classical topology. Let $\pi {\colon}E\to S$ be a proper fiber bundle of topological manifolds whose fibers carry a suitable $n$-dimensional orientation. Then there is a map $\pi _*{\colon}h^{\bullet }(E)\to h^{\bullet -n}(S)$ variously termed the ‘pushforward’ or ‘direct image’ or ‘wrong-way map’ or ‘*umkehr* map’ or ‘Gysin map’ or ‘transfer’. The analogy with our situation is tighter if we think of this pushforward on the level of cochains, or geometric representatives, rather than cohomology classes.

Formulas  and , together with a similar formula for compact surfaces with boundary, can be used to define the functor $F$. This constructs a 1-2 theory. We would like, however, to continue down to points, i.e., to a 0-1-2 theory. To express the higher gluing laws encoded in an $n$-dimensional 