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Results tagged with gt.geometric-topology
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user 1358
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
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What kind of 3-manifolds arise has hypersurfaces in R^4?
What kind of 3-manifolds can arise as hypersurfaces $\{ f(x,y,z,w) = 0\} \subset \mathbb{R}^4$? Can they have nontrivial H1 or H2?
- 22.8k
3
votes
1
answer
341
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What is being braided in SL(2,Z)?
The braid group on 3 strands is a central extension of the modular group. By definition,
\[ B_3 = \langle \sigma_1, \sigma_2: \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2 \rangle \]
This group h …
- 22.8k
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answers
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$ABA^{-1}B^{-1} = E$ the topology of the space of non-commuting matrices
Is there any discussion of topology of space of matrices
$ ABA^{-1}B^{-1} = E $ with $E$ diagonal matrix, $A,B \in \mathrm{GL}(n,\mathbb{C})$?
E.g. is this a variety of just a scheme? How many com …
- 22.8k
2
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1
answer
164
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the space of noncrossing partitions of S^1
A non-crossing partition of the set $\mathbb{Z}_{mn}=\{ 1, 2,\dots,m n\}=\bigcup X_i$ is a disjoint union of sets such that
if $ a < b \in X_1$ and $ c < d \in X_2$ then we can't have $a < c < b < …
- 22.8k
13
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2
answers
791
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"C choose k" where C is topological space
One day I read a generating function in a paper. For any "sufficietly nice topological space", $C$:
$$ \sum_{l \geq 0 } q^{2l}\chi(\mathrm{Sym}^l[C]) = (1 - q^2)^{-\chi(C)} = \sum_{l \geq 0} \binom{ …
- 22.8k
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1
answer
521
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Can vector fields be used to construct diffeomorphisms of the 2-sphere? [closed]
For some reason, today I want to understand better the group of diffeomorphisms of the 2-sphere, $S^2$.
After a few minutes I found this result by Smale in 1958.
The space $\Omega$ of all orien …
- 22.8k
5
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1
answer
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Examples of Morse functions on unit tangent bundle of the sphere $T^1(S^2)$
In a sense, calculus is all about the study of critical points of functions on flat space $\mathbb{R}^N$ (e.g. here). Let's try a different venue, the unit tangent bundle of the sphere.
$$ T^1(S^2) …
- 22.8k
5
votes
0
answers
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deformed Gauss Bonnet formula?
I was reading about Gauss-Bonnet for circles, that integrating the curvature over the circle $\gamma(t)$ leads to the number of times $\gamma'(t)$ rotates around the origin. (The degree of the Gauss …
- 22.8k
13
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1
answer
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Is there a topograph for Pythagorean triples?
I have been reading Allen Hatcher's notes on quadratic forms. Naturally, we draw a picture encoding all the values of a quadratic form in a topograph. These are build by iterating the parallelogram …
- 22.8k
11
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2
answers
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Thurston-Cannon $S^2$-filling curves
I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers:
Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry
Gro …
- 22.8k
2
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1
answer
219
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examples of surface diffeomorphism that exhibit heteroclinic bifurcation?
I was reading about horseshoes and heterclinic bifurcation but my knowledge of dynamical systems is really old fashioned.
as I understand the local stable manifold and the local unstable manifold in …
- 22.8k
3
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Seifert-Fibered 3-Manifolds and Rotation Numbers
I was trying to understand how the "ziggurats" come about in the paper by Calegari and Walker.
Motivating Question Given a free group F, and an element w of F, and given
values of the rotation …
- 22.8k
2
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2
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588
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Hausdorff Dimensions of Limit set of subgroups of SL(2,Z)
In a recent paper by Bourgain, Sarnak, Gamburd [1] talks about subgroups of $SL(2,\mathbb{Z})$.
Let $\Lambda$ be a finitely generated non-elementary subgroup of $SL(2,\mathbb{Z})$ with Hausdorff …
- 22.8k
1
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0
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square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces
I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant: …
- 22.8k
4
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1
answer
392
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braids and dynamics of roots of a polynomial
The 2-variable polynomial equation $f(z,t) = 0$ with $z = \mathbb{C}, \\,t \in \mathbb{S^1}$ has $n = \mathrm{deg}_z f$ solutions each fixed $t$. I wanted to follow the roots as they travel with time …
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