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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.

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. Main branches of algebraic topology at least include 1 Homotopy groups 2 Homology 3 Cohomology 4 Manifolds 5 Knot theory 6 Complexes

17 questions
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Simple explanation for what a torsor is

I am studying Chris Elliott's notes on Line and Surface Operators in Gauge Theories (available here). In the notes, there's a mention of the fact that (for $G = U(1)$), $$W_{\gamma, n}(A) = e^{in\oint_{\gamma}A}.$$ The gauge field $A$ is not…
leastaction
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Classification of higher Symmetry Protected Topological (SPT) phases

Suppose that we have a $d$ dimensional bosonic SPT phase, protected by some $p$-form symmetry, $G^{[p]}$. Suppose also that it is classified within group cohomology, so that we don't have to run into cobordisms. It is not clear to me what type of…
ɪdɪət strəʊlə
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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 the degeneration of certain functions on…
JaP
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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 continuous deformation of spacetime. I find these to be…
D E
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Physical meaning of the Yang-Baxter equation

I'm a graduate student in mathematics, and I have lately been interested in the relation between knot theory and statistical mechanics. As I understood, the Yang-Baxter equation (shown below) is the equivalent of the Reidemeister III move (RIII),…
Léo S.
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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 (theoretical) physics, and he told me that indeed…
George Smyridis
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What is the topology of non-entangled states region for a 2 qubit Bloch hypersphere?

Preamble A two qubit/spin-1/2 system can be represented…
Mauricio
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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}^s{\frac{N_I}{2\pi} B^I \wedge dA^I} +…
PhysicsMath
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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 quarks or in the mesons or even approximately in…
wonderich
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Reidemeister Torsion

Can somebody in a layman's language explain what is a Reidemeister Torsion? This seems to play an important role in path integral of 2+1-gravity as demonstrated here in arXiv:gr-qc/9406006. This is the only place I know in physics where algebraic…
Dr. user44690
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String topology in string theory

How do string topology, string field theory and topological strings interact? Does anybody see a global picture? By string topology I mean the TQFT based on the homology of the space of loops described in https://arxiv.org/abs/0711.4859
Pavel
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Winding invariant for non-contractible loops

I am reading this paper where I am confused about equation 16. It follows that (one-dimensional) loops can be linked with $K_0$ or, in other words, the space of gapped Floquet evolutions $SU (N )\backslash K_0$ has noncontractible loops, with…
wooohooo
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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: A finite group of anyons has some character that…
programjames
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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 light travelling through the tunnel. So, in…
Robert
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Which geometry does not allow the existence of matter?

I have seen these lectures by Fredric Schuller that discuss the obstruction theory and the role of global geometric properties in admitting a spin structure. See the video at 01:27:52 https://youtu.be/Way8FfcMpf0?t=5279 For example, it is said that…
VVM
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