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This is too big a question to answer fully. Let me start by explaining why in a single symmetry class we need so many invariants. Consider just Chern insulators, and ignore interactions. We want to work with crystals and quasicrystals and amorphous systems, for both infinite and finite area systems. We want to work with various boundaries and boundary ...

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Question 1: what does orientable manifold has to do with spinor? Spinor fields cannot be consistently defined on an arbitrary manifold. As the Encyclopedia of Mathematics explains, “Necessary and sufficient conditions for the existence of a spinor structure on $M$ consist of the orientability of $M$ and the vanishing of the Stiefel–Whitney class $W_2(M)$. ...

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It's possible that the universe has the form of a curved, spherelike 3d space (you can say curved in a 4d Euclidean space, but in General Relativity, space is inherently curved, inherently curved, hand in hand with time). Like a curved, spherelike 2d sphere, as mentioned in your question. Observations suggest that the visible universe is flat. So space ...

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Singularity: Suppose we have the unit disc in $\mathbb R^2$ with a singularity at $r=0$. Then we would remove the point at $r=0$ and get the punctured unit disc $0<r<1$. Now we can apply a well-behaved (no singularities) coordinate transform $r\mapsto r+1$ to get a ring $(1,2)\times S_1$. This transforms the singularity from a point to the circle $r=... 2 These arguments are consistent with each other. The difference is the kind of field they handle. Since the arguments are about instantons, let's forget about$\theta\$-terms, which are just a distraction. An instanton is a field configuration which is a local minimum of the action. The reason topology comes up when thinking about instantons is that is that ...

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