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Question came up when reading this answer. How is it possible that we can choose different topological spaces to model a same physical scenario?

If we have such different spaces, so many things will be different. For example, convergent sequences, what points there can be and qualitative features like holes. So wouldn't these features mess up our model?

Qmechanic
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tryst with freedom
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  • Because there are non-trivial diffeomorphisms? That is a feature of the mathematics, not one of physics, though. Physical reality doesn't change just because we can describe it in different coordinate systems. Physics is all that which is invariant under such transformations. – FlatterMann Oct 25 '22 at 06:22
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    Diffeomorpjisms is not exactly about coordinate systems. The shift in coordinates are jn the chart transition maps not in diff manifold level @FlatterMann – tryst with freedom Oct 25 '22 at 06:28
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    It is still correct that physics is not the same as the description of physics. The physics of a system is unique, but there is an infinity of different descriptions for it. That is a redundancy that only exists in the mathematics. On the physical level two cars either collide in the intersection or they don't. How you describe that does not matter. – FlatterMann Oct 25 '22 at 06:38
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    @TrystwithFreedom sorry I do not get the question. The answer you linked just says that you can study gravity (GR or Newton gravity) over different base manifolds.. that's all. Similarly, you may study heat conduct on a sphere or on a torus, or the plane... What's the question? – Quillo Oct 25 '22 at 07:22
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    The choice of models effect s the features of space in consideration. So, are there no properties of motion related to feature of space? Is Motion completely independent of the arena it takes place in?

    Hope it makes more sense now @Quillo

     – tryst with freedom Oct 25 '22 at 07:23
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    @TrystwithFreedom now it's clear, thank you. Locally the equations of motion EOM are the same (for manifolds that are locally flat). But the EOM are not the only thing that define the evolution (boundary conditions and initial conditions are also needed, these depend on the global topology). It may be instructive to start with the heat equation on the infinite plane VS heat on a torus to get a feeling of the difference. Ofc different topologies do not refer to the same physical situation (it's different to live on a sphere rather than on a plane). – Quillo Oct 25 '22 at 07:36
  • @TrystwithFreedom I don't see how that answer says that you can have different topological spaces for the same physical scenario? – MBN Oct 26 '22 at 10:59
  • See the section near topological censorship conjecture @MBN – tryst with freedom Oct 26 '22 at 11:09
  • @TrystwithFreedom : It says the opposite! "Depends on what you're doing and the conditions of the physical problem." You can have different topologies for diiferent physical problems, not for the same scenario. – MBN Oct 27 '22 at 09:52

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A (topological) manifold $(\mathcal M,\mathcal O)$ is locally homeomorphic to $\mathbb R^n$, which we implicitly take to be equipped with the standard topology. So already, local topological questions like topological completeness are answered in the affirmative.

The global topology of a manifold is another story - so if you mean holes in the sense of singular homology, then indeed you have an infinity of possible choices you could make (possibly subject to additional constraints you wish to impose for physical, mathematical, or philosophical reasons). Generically such features would have an observable effect, so it is then a matter of choosing the model which fits best with your observations.

J. Murray
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  • Oh okay, so the resolution is that locally speaking, all the topological manifolds are equivalent to each other. – tryst with freedom Oct 25 '22 at 07:44
  • https://physics.stackexchange.com/questions/54531/does-local-physics-depend-on-global-topology – tryst with freedom Oct 26 '22 at 11:12
  • @TrystwithFreedom Do you have a follow-up question? – J. Murray Oct 26 '22 at 14:12
  • https://physics.stackexchange.com/questions/733848/can-we-just-take-the-underlying-set-of-the-space-time-manifold-as-mathbbr4 – tryst with freedom Oct 26 '22 at 14:12
  • @TrystwithFreedom I said that a manifold is locally homeomorphic to $\mathbb R^d$. That means that for each point $p$ we can find a small neighborhood of $p$ which is homeomorphic to a small patch of $\mathbb R^d$. It certainly does not imply that all $d$-dimensional manifolds are homeomorphic to a subset of $\mathbb R^d$. – J. Murray Oct 26 '22 at 14:19
  • @TrystwithFreedom It is true that any $d$-dimensional manifold can be understood as a subset of $\mathbb R^n$ for suitably high $n$ via the Whitney embedding theorem, but differential geometry allows us to talk about the intrinsic properties of a manifold without referencing some higher-dimensional space in which it is embedded. – J. Murray Oct 26 '22 at 14:23