  We connect topological changes that can occur in $3$-space via surgery, with
black hole formation, the formation of wormholes and new generalizations of
these phenomena, including relationships between quantum entanglement and
wormhole formation. By considering the initial manifold as the $3$-dimensional
spatial section of spacetime, we describe the changes of topology occurring in
these processes by determining the resulting $3$-manifold and its fundamental
group. As these global changes are induced by local processes, we use the local
form of Morse functions to provide an algebraic formulation of their temporal
evolution and propose a potential energy function which, in some cases, could
give rise to the local forces related to surgery. We further show how this
topological perspective gives new insight for natural phenomena exhibiting
surgery, in all dimensions, while emphasizing the $3$-dimensional case, which
describes cosmic phenomena. This work makes new bridges between topology and
natural sciences and creates a platform for exploring geometrical physics.
