space $X$ with projection $p_X : X \times [0,1] {\longrightarrow}X$. This shows that ${p_{\partial M}}_* (\partial\nu \times [0,1])$ is a cycle. For dimensional reasons it must be a boundary as well. This shows that vanishes and finishes the proof.

The Main Theorem
----------------

We can now state our main result:

\[mainthm\] For any topological space $T$ and any cocycle $\theta \in Z^n(T;{\operatorname{U}}(1))$ $$\begin{aligned}
 T_\theta : {T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)}{\longrightarrow}{{2\mathbf{Vect}_{\mathbb{C}}}}\end{aligned}$$ is an invertible homotopy quantum field theory with target $T$.

Thanks to the Lemmata \[lemmacompof1mor\], \[lemmaverticalcomp\] and \[lemmahomotopyinvariance\] it remains to prove the following:

-   Horizontal composition: Given 1-morphisms $$\begin{aligned}
            (\Sigmait_a,\varphi_a) : (S_0,\xi_0) &{\longrightarrow}(S_1,\xi_1),\\
            (\Sigmait_b,\varphi_b) : (S_1,\xi_1) &{\longrightarrow}(S_2,\xi_2)
            \end{aligned}$$ and 2-morphisms $$\begin{aligned}
            (M,\psi) :  (\Sigmait_a,\varphi_a) &\Longrightarrow     (\Sigmait_b,\varphi_b),\\
                    (M',\psi') :    (\Sigmait_b,\varphi_c) &\Longrightarrow     (\Sigmait_c,\varphi_c)
            \end{aligned}$$ we have to show that for fundamental cycles $\sigma_0$, $\sigma_1$ and $\sigma_2$ of $S_0$, $S_1$ and $S_2$, respectively, the square $$\begin{tikzcd}
            \Sigmait_a^{\varphi_a} (\sigma_1,\sigma_0)\otimes \Sigmait_b^{\varphi_b} (\sigma_2,\sigma_1) \ar{rr}{{\Phiit}} \ar[swap]{dd}{T_\theta (M)_{\sigma_1,\sigma_0}\otimes T_\theta (M')_{\sigma_2,\sigma_1}} & & (\Sigmait_b \circ \Sigmait_a)^{\varphi_b\cup \varphi_a} (\sigma_2,\sigma_1) \ar{dd}{T_\theta (M' \circ M)_{\sigma_2,\sigma_0}} \\
            & & \\
            \Sigmait_b^{\varphi_b} (\sigma_1,\sigma_0)\otimes \Sigmait_c^{\varphi_c} (\sigma_2,\sigma_1) \ar{rr}{{\Phiit}} & & (\Sigmait_c \circ \Sigmait_b)^{\varphi_c\cup \varphi_b} (\sigma_2,\sigma_1)
            \end{tikzcd}$$ featuring as the horizontal arrows the isomorphisms from the proof of Lemma \[lemmacompof1mor\] commutes. By picking representatives for the fundamental classes as in 