New answers tagged gt.geometric-topology
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Fundamental group of a generalized connected sum
Disclaimer: I wrote this answer before realizing the question was specifically about codimension 1. This answer only works in codimension 1 for the case that the normal bundle is not orientable.
First,...
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Fundamental group of a generalized connected sum
You want to look up the idea of “free product with amalgamation”. Also, you need side-conditions on how $S$ lies in the manifolds $M$ and $N$ - namely, the normal bundles need to be isomorphic. ...
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For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?
More generally, for any closed subset $S$ of a complete manifold $M$, the set of points $x$ at whose minimal distance to $S$ is attained at more than point has measure $0$.
Indeed, consider the ...
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Linking number and intersection number
$\DeclareMathOperator\tX{\widetilde{X}}\DeclareMathOperator\tB{\widetilde{B}}\DeclareMathOperator\tD{\widetilde{D}}\DeclareMathOperator\Z{\mathbb{Z}}$
In fact, $B$ must intersect $D$ at least $|\text{...
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The boundary regularity of a Teichmüller domain
To follow up on Moishe Kohan's comment (and my own):
Bromberg conjectures, in his 2011 paper The space of kleinian punctured torus groups is not locally connected, that the same holds for any surface $...
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5
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Prove these are not surface groups
Here's an argument that uses only basic combinatorial group theory (Reidemeister-Schreier).
Let $n \geq 2$, and let $G = \langle a_1, b_1, \dots, a_g, b_g | [a_1, b_1]^n [a_2, b_2] \cdots [a_g, b_g] \...
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Prove these are not surface groups
Here is an answer that is inspired by but not using Gromov's simplicial norm considerations. If we use the Gromov norm, we can distinguish all $\Gamma_{g,n}$; See the second part of the answer.
The ...
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4
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Relationship between quotient CW-complexes after attaching cells
If I understand the question correctly, you have a CW complex $Y'$ which is the union of two subcomplexes $Y$ and $X'$ whose intersection is the subcomplex $X$. We can first collapse $X$ to a point ...
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6
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Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
This fact doesn’t need $M$ to be hyperbolic. It just needs one general theorem about 3-manifold topology, namely the Scott core theorem.
Let $N\to M$ be the covering space corresponding to the ...
- 24.1k
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Continuous extensions of tangent vector fields
I claim that $F$ exists if and only if $G$ has zero winding number around the boundary of $\Omega$. I will only give a complete proof of one direction.
We can suppose that $\Omega$ is a plane domain, ...
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Continuous modification of tangent vector fields
Think of $\Omega$ as an open set in the plane. If $F$, restricted to the boundary, has a different winding number than $G$, then all deformations of $F$ through unit vector fields still have that same ...
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5
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Residual finiteness and a gluing problem
Thurston never finished his project, hence, we cannot know for sure what exactly did he have in mind in this part of the diagram. Here is what we know:
Fundamental groups of good compact 3-...
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Confusion about Teichmüller curves and $\operatorname{SL}_2$-action
Veech coined the term "Teichmueller curve" and McMullen popularized it. A Teichmueller curve is a curve in the classical Riemann moduli space that is totally geodesic with respect to the ...
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Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle over a surface
Why is the Euler number of a Seifert bundle a "natural" generalization of a circle bundle over a surface?
I can see at least three reasons:
A circle bundle is a Seifert manifold with no ...
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8
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Loop manipulation subgroup of the braid group
Your group $H_n$ is (I believe) called the wicket group by Brendle and Hatcher in their paper Configuration spaces of rings and wickets. They provide a presentation in Proposition 3.6. They also ...
- 26.4k
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Dehn surgery on $RP^2 \times S^1$
The boundary of your solid torus $T_L \cong D^2 \times S^1$ is equipped with a pair of foliations by circles (at "right angles"). The first one is the foliation by circles of the form $\...
- 26.4k
15
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Identifying two definitions of orientation on a vector space
Here's a direct way to relate the two:
One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing
$$
\Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n+...
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Residual finiteness of hyperbolic 3-manifold groups
The answer to Q1 is negative in general (allowing infinitely generated fundamental group). See Example 2 which is a discrete torsion-free subgroup $G< PSL_2(\mathbb{C})$, hence $\mathbb{H}^3/G$ is ...
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0
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Simple convergence of convex compact set implies Hausdorff convergence
A counterexample is given by $n=2$, $C=\{(0,0)\}$, and $C_k=\{(t,kt)\colon0\le t\le1\}$ (or one can instead take $C_k=\{(t,t/k)\colon0\le t\le1\}$).
Another counterexample, in the same spirit, is ...
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Residual finiteness of hyperbolic 3-manifold groups
Here's another negative answer for Q2. I'm assuming (as in Sam Nead's answer) that the covering should be locally isometric. By Ahlfors-Bers, a tame infinite volume hyperbolic manifold with ends of ...
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Residual finiteness of hyperbolic 3-manifold groups
Sam Nead's answer does it, but perhaps I can offer a slightly different perspective on Question 1. No complicated hyperbolic gluing results are needed.
I assume we are satisfied with the ...
- 24.1k
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Residual finiteness of hyperbolic 3-manifold groups
The answer to the first question is "yes" and the answer to the second is "no", assuming that you are looking for a covering which is a locally isometric. If you do not require a ...
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