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Well, one of the "mantras" of General Relativity is:

Einstein Field Equations concerns about the local geometrical structure of spacetime (the metric tensor) and tell you nothing about the fixed undelying topology of spacetime.

One way to study this fact is in the realm of Spherically Symmetric Wormholes. Specifically, when we consider both Black Hole and White Hole solutions and a kruskal local chart, we can produce (at Kruskal time $v = 0$) a non-traversable wormhole known as the Einstein-Rosen Bridge. Furthermore, we have Morris-Thorne Traversable solution which acts as a remedy to the disease of non-traversability of Einstein-Rosen Bridges.

In $[1]$, Thorne works out the Schwarzschild Solution and the realization of the Einstein-Rosen bridge (or Kruskal Throat) and then show us two figures:

enter image description here

In fact he then wrote:

"(b) An embedding of the Schwarzschild space geometry at "time" $v = 0$, which is geometrically identical to the embedding (a), but which is topologically different."

My difficult here is how can I show formally the differences between $(a)$ and $(b)$. I mean, to construct $(a)$ you must to use an specific embbeding procedure: using cilindrical coordinates. But the whole discussion here is about topology. Then I would like to ask:

What is the topology of (b)?

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$[1]$ Misner.C, Thorne.K, Wheeler.J. Gravitation. pages 836-837.

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  • $\begingroup$ " I mean, to construct (a) you must to use an specific embbeding procedure: using cilindrical coordinates" Any embedding must nescessarily preserve topology and it is usually much easier to see topology directly from picture (abstracting the additional, nontopological structures away in your head) than to talk about topology directly in fully abstract way. $\endgroup$ – Umaxo Nov 12 '20 at 13:20
  • $\begingroup$ @Umaxo Thanks for your answer. But, the thing is, if someone asks for a mathematical explanation on the difference between the undelying topology I won't be able to give. For example, I'm asking for something like: "Hey friend, what suppose to mean Euclidean Topology?" "Well mate, it means precisely the topology generated by the balls $B_{r}(p) := \{ x\in \mathbb{R}^{n} \mid d(p,x) < r \}$. For my question,"Hey friend, what suppose to mean the Topology of Wormhole $(b)$?" "Well mate, it means precisely the topology generated by.....". I need a mathematical expression, than just a draw. $\endgroup$ – M.N.Raia Nov 12 '20 at 13:39
  • $\begingroup$ How about giving the basis by intersection of embedded surface with open balls in manifold in which it is embedded? $\endgroup$ – Umaxo Nov 12 '20 at 13:46
  • $\begingroup$ This does not prove that the two topologies are not homomorphic though. $\endgroup$ – Umaxo Nov 12 '20 at 13:47
  • $\begingroup$ As an hint, I suggest using Euler characteristic (in some generalised framework rather than polyhedrals) $\endgroup$ – Pipe Nov 13 '20 at 20:20
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You need to compare the topological data of these two manifolds. In this particular example they have different homotopy group, i.e. in the second example you do not need to cross the throat to get into the other side but in the first one you do need cross the throat.

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