# Questions tagged [algebraic-topology]

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.

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### Are hierarchal states/fractional exchange statistics equivalent in the FQHE?

There are several theories for the fractional quantum Hall effect. The last listed in the Wikipedia article are composite fermions, though these seem to be a subset of fractional exchange statistics: ...
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### How is a wormhole (Einstein-Rosen Bridge) different than a tunnel?

What is the difference between an Einstein-Rosen Bridge (wormhole) and a tunnel through a mountain? Obviously, light that travelled around the mountain would take longer to reach other side so that ...
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### Can topological invariants be related to Noetherian charges?

I recently attended a seminar on physical mathematics, and learned about some topological invariants, especially in 4D spaces. These topological invariants are considered to be invariant under ...
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### What is the correct domain of integration for the index of instantons? - $\mathbb{R}^4$ or $S^4$?

I posted the original question on Math SE but it seems like a more appropriate question for Physics SE: https://math.stackexchange.com/q/4417225/ In calculating the instanton solutions for $SU(2)$ ...
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### Utility of Topological Data Analysis in Theoretical Physics

I audited a lecture on Topological Data Analysis, and I found it really interesting, primarily because of the connection to algebraic topology. I asked the professor if there are any connections to (...
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### Large gauge transformation and intersection form

I am reading this paper and on pp.19-20 it states the following relation between large gauge transformation and intersection form: for the action on a 4-manifold $M^4$ S[A,B] = \int_{M^4}{\sum_{I=1}...
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### Physical application of Postnikov tower, String$(n)$ and Fivebrane$(n)$
We know that the Spin group is quite a useful concept in physics. For example, Spin$(3)=SU(2)$ (and Spin$(6)=SU(4)$) that describe gauge groups in the Standard model and the isospin symmetry in the ...
### What is the importance of studying degeneration on $M_g$
Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study ...