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6 votes
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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{...
Andy Putman's user avatar
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1 vote
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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 $...
Sam Nead's user avatar
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5 votes

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] \...
Carl-Fredrik Nyberg Brodda's user avatar
3 votes

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 ...
Lvzhou Chen's user avatar
4 votes
Accepted

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 ...
Allen Hatcher's user avatar
6 votes
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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 ...
HJRW's user avatar
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2 votes
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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, ...
Ben McKay's user avatar
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2 votes
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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 ...
Ben McKay's user avatar
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5 votes
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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-...
Moishe Kohan's user avatar
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2 votes

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 ...
Chris Judge's user avatar
5 votes

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 ...
Marco Golla's user avatar
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8 votes
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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 ...
Sam Nead's user avatar
  • 26.3k
4 votes

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 $\...
Sam Nead's user avatar
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15 votes
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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+...
Achim Krause's user avatar
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3 votes

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 ...
Ian Agol's user avatar
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0 votes
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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 ...
Iosif Pinelis's user avatar
6 votes

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 ...
Danny Ruberman's user avatar
5 votes

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 ...
HJRW's user avatar
  • 24.1k
4 votes

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 ...
Sam Nead's user avatar
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8 votes
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Hyperbolic three-manifolds that fiber over the circle

Question: Is the length of homotopically non-trivial loops in $M(f)$ bounded below in terms of the genus $g$ of the fiber? Answer: No. Here is one family of examples. Suppose that $S$ is the closed ...
Sam Nead's user avatar
  • 26.3k
1 vote

Space of the trivial long knot in the thickened surface

Let us show that $\mathcal E=Emb_0(I,F\times I)\sim\Omega_0(F,x_0)$. We start with R. Budney's remark. Proposition. Let $F$ be a connected compact 2-manifold and $P(F)$ the pseudoisotopy group, i.e ...
nim's user avatar
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