surface $\Sigma$ of $M$ such that $U_\pm \subseteq \Sigma_\pm : = I^\pm_M(\Sigma)$. Hence, there exists a morphism $(\iota_{{\underline{U}}}^M: {\underline{U}}\to M)\to (\iota_{{\underline{\Sigma}}}^M : {\underline{\Sigma}}\to M)$. As a consequence, our original problem reduces to finding a zig-zag of morphisms in $\PP_M$ between $\iota_{{\underline{\Sigma}}}^M : {\underline{\Sigma}}\to M$ and $\iota_{{\underline{\Sigma}}^\prime}^M : {\underline{\Sigma}}^\prime \to M$, for any two Cauchy surfaces $\Sigma,\Sigma^\prime$ of $M$. To exhibit such a zig-zag, let us introduce $\widetilde{U}_+ := \Sigma_+ \cap \Sigma^\prime_+$ and $\widetilde{U}_- := \Sigma_- \cap \Sigma^\prime_-$. If we could prove that $\iota_{\widetilde{U}_\pm}^M : \widetilde{U}_\pm \stackrel{c}{\to} M$ are Cauchy morphisms, then

(\_\^M : M) (\_\^M : M) (\_[\^]{}\^M : \^M)

would provide a zig-zag that proves connectedness of $\PP_M$.

It remains to show that $\widetilde{U}_+ = \Sigma_+ \cap \Sigma^\prime_+ = 
I^+_M(\Sigma) \cap I^+_M(\Sigma^\prime)\subseteq M$ contains a Cauchy surface of $M$. (A similar argument shows that $\widetilde{U}_-\subseteq M$ also contains a Cauchy surface of $M$.) Because $\Sigma,\Sigma^\prime$ are by hypothesis Cauchy surfaces of $M$, there exists a Cauchy surface $\Sigma_1\subset I^+_M(\Sigma)$ of $M$ in the future of $\Sigma$ and a Cauchy surface $\Sigma_1^\prime \subset I^+_M(\Sigma^\prime)$ of $M$ in the future of $\Sigma^\prime$. We define the subset

\[eqn:widetildeSigma\] := (\_1 J\^+\_M(\_1\^)) (J\^+\_M(\_1)\_1\^) \_+ M

and claim that $\widetilde{\Sigma}$ is a Cauchy surface 