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Generally in textbooks they represent spacetime as $(M,\nabla,g,t)$ where $M$ is a Lorentzian manifold,$\nabla$ a torsion-free connection,$g$ a metric and $t$ a time orientation. But they do not talk of topology. My question is does the topology of spacetime have importance in physics?

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    $\begingroup$ As with any manifold the open sets defined by the topology are used in defining charts. $\endgroup$
    – Charlie
    Sep 19, 2020 at 16:04
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    $\begingroup$ Usually $\nabla$ is the Levi-Civita connection associated to $g$. However, the topology is included in the definition of the smooth manifold $M$. $\endgroup$ Sep 19, 2020 at 16:07
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    $\begingroup$ The second Stiefel-Whitney class has to vanish so one can define spinors on the manifold. And obviously this last has to be orientable. $\endgroup$ Sep 19, 2020 at 16:24
  • $\begingroup$ You cannot meaningfully define a smooth manifold without looking at the topology. As an example it $M$ needs to be second-countable which is a topological property. $\endgroup$ Sep 19, 2020 at 16:32
  • $\begingroup$ More on the topology of spacetime. $\endgroup$
    – Qmechanic
    Sep 19, 2020 at 17:27

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Spacetime topology certainly has importance in physics.

In condensed matter physics, topological phases of matter are systems which are sensitive to spacetime topology. Many of the interesting ones are two dimensional, so they can in principle be fabricated into interesting surfaces in a lab. Alternatively, through clever tuning of non-local interactions, physics on non-trivial 3-manifolds can be engineered as well.

In cosmology there is the question of the topology of the universe. It was once hypothesized that the universe is a Poincare homology sphere. Wormholes give the universe interesting topology, but that topology is usually hidden behind an event horizon. This general phenomenon is known as cosmic censorship, and is an active area of study.

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  • $\begingroup$ Aren't there also particle models that depend on non-trivial topologies? $\endgroup$
    – S. McGrew
    Sep 19, 2020 at 21:07
  • $\begingroup$ @S.McGrew What do you have in mind? $\endgroup$ Sep 19, 2020 at 22:34
  • $\begingroup$ For example, the electron model proposed by Hestenes: vixra.org/pdf/1408.0203v2.pdf. $\endgroup$
    – S. McGrew
    Sep 20, 2020 at 0:47

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