$$\frac{1}{\operatorname{vol}G}\int_{G} \lambda (g)\,dg =\begin{cases} 0,&\lambda
     \not= 1;\\1,&\lambda =1.\end{cases}$$

\[thm:4\] There are cohomology theories $h$ with the property that $h^2(BG)$ is isomorphic to the set of ${{{{\mathbb Z}}/2{{\mathbb Z}}}}$-graded abelian characters $(\lambda ,\epsilon
)$, i.e., $\lambda {\colon}G\to {\mathbb T}$ is an abelian character and $\epsilon =0,1$. For example, we can take cohomology theory $h$ determined by a spectrum with two nontrivial homotopy groups: ${{\mathbb Z}}$ in degree 0 and ${{{{\mathbb Z}}/2{{\mathbb Z}}}}$ in degree $-2$ connected by the nontrivial $k$-invariant. A less efficient alternative is the Anderson dual of the sphere [@HS Definition B.2]. In either case spin structures are required to define pushforward, so the corresponding field theory is defined on spin manifolds. It is then natural to let the values of the theory lie in ${{{{\mathbb Z}}/2{{\mathbb Z}}}}$-graded complex vector spaces. If $\epsilon
=0$, the theory factors through oriented manifolds and reduces to the previous. If $\epsilon =1$, then the classical theory assigns the odd line ${{\mathbb C}}$ to the unique connection on a point. For $\lambda \not= 1$ we obtain the trivial theory as above, but now if $\lambda =1$, then $F({\ensuremath{S^1}}_{\textnormal{n}})=-1$ and $F({\ensuremath{S^1}}_{\textnormal{b}})=1$, where the subscript indicates whether we consider the circle as endowed with the nonbounding 