In the ADM formulation of general relativity it is assumed that the spacetime topology is $\Bbb{R}\times \Sigma$. Suppose I wanted to consider spacetimes that undergo topology change with foliation time, would it be possible to describe such dynamics within some canonical formulation? I have been searching for an extension of the ADM formalism that lifts the topological restriction, but I can't seem to find anything on the subject (there are a few papers describing topology change in classical GR in the covariant formalism but that is all I could find).
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$\begingroup$ You can look up the more general formalisms of the De Donder-Weyl/polysymplectic/multisymplectic formalism, which does not assume anything about the underlying space. $\endgroup$– SlereahCommented Dec 8, 2022 at 21:24
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In the ADM formulation the topology is not assumed but deduced. Thanks to Geroch [1] theorem, if spacetime is globally hyperbolic then topologically it is $\mathbb{R} \times \Sigma$ where the topology of $\Sigma$ do not change in time. The idea of the proof is as follows: consider two leaves of the foliation, $\lbrace t_{0} \rbrace \times \Sigma_{0}$ and $\lbrace t_{1} \rbrace \times \Sigma_{1}$. If spacetime admits everywhere a Lorentzian metric (and is orientable and time orientable) then there exist a non-vanishing vector field all over spacetime. Then we can use the integral curve of this vector field to map $\Sigma_{0}$ into $\Sigma_{1}$ showing they must have the same topology.
[1] https://pubs.aip.org/aip/jmp/article/8/4/782/460173/Topology-in-General-Relativity
