15 votes
Accepted

Does a compact contractible metric space have a point that is fixed by all isometries?

There are finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the Mostow-Palais embedding theorem, which equivariantly and ...
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10 votes
Accepted

Coefficient of the top Pontryagin class in $L$-genus

The coefficient of $p_k$ is given by $$2^{2n}(2^{2n-1}-1)\frac{B_n}{(2n)!} = \zeta(2n)\frac{2^{2n}-2}{\pi^{2n}},$$ see e.g. Appendix A of this older version of Weiss (warning: for Weiss, the ...
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  • 7,432
7 votes
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How wild can an open topological 3-manifold be if it has a compact quotient?

The possible universal covers of closed 3-manifolds are $S^3-C$, where $|C|=0, 1, 2$ or $C$ is a tame Cantor set, corresponding to the space of ends of the fundamental group as you suspect. This ...
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  • 61.2k
6 votes
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Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?

Given a knot in $S^3$, think of it as an embedding $$f : S^1 \to S^3.$$ The configuration space of $5$ distinct points in $S^3$ is denoted $C_5(S^3)$, this is a $15$-dimensional manifold and it ...
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  • 40.6k
5 votes
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Symmetries of contractable subsets of $\Bbb R^n$

I posted a refined/generalized question Does a compact contractible metric space have a point that is fixed by all isometries? and received an answer that contained all essential ingredients to ...
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  • 9,471
5 votes
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Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies?

I think your questions are answered by this paper of Leininger, McReynolds, Neumann and Reid. In their terminology, your first question is asking about manifolds with equal length sets, and your ...
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  • 22.3k
4 votes
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Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?

When the manifold is the universal cover of a compact $3$-manifold $M$ (to begin with, lets say without boundary) then you construct the embedding by hands, using geometrization. In your question let'...
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  • 40.6k
4 votes
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A ribbon presentation for a torus knot

As this composite knot is an example of a symmetric union of knots I suggest that you read my paper "The search for nonsymmetric ribbon knots" (see for instance Figures 1 and 4 and the ...
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4 votes

Reference for an easy lemma on homeomorphisms of connected manifolds

In [Ancel and Bellamy, On homogeneous locally conical spaces, Fund. Math. 241 (2018), no. 1, 1–15] it is shown that every homogeneous locally conical connected separable metric space that is not a $1$-...
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4 votes
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Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?

Ian Agol, in the comments says: Yes, there should be plenty. Think of the Seifert surface for the 5_2 knot as a disk with two strips (1-handles) attached. By tying knots into the strips (with zero ...
3 votes

Alexander invariant of torus knot

Putting everything together, and with the big help of "Knots" by Burde, I found the following solution. Let $f_n(t)\colon= 1+x+\cdots+x^{n-1}$, we note that $\frac{t^m-1}{t-1}=f_m(t)$. @...
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3 votes

Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?

Maybe a different approach but having the same flavor: Take a (untwisted) Whitehead double of any nontrivial knot $K$. Denote such a double knot $WD(K)$. Construct a satellite knot using $5_2$ knot as ...
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  • 783
3 votes
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Is every finitely generated classical Schottky group quasifuchsian?

Yes - every Schottky group is quasi-fuchsian. See Lemma 1 of Chuckrow's paper "On Schottky Groups with Applications to Kleinian Groups" published in Annals of Mathematics, 1968. The ...
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  • 18.5k
2 votes

Local complexity of triangulations

Theorem 3.37 of Florian Frick's thesis gives a sketch of the proof of statement in the question for smooth manifolds. At first sight, it only uses the Whitney embedding theorem and transversality, ...
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  • 7,432
2 votes
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Complex length of geodesic added in hyperbolic Dehn surgery

After more thought, I realize this can't possibly be true, by an easy extension of the methods in [NZ]. We can re-write our two equations $pu + qv = 2\pi i$ and $\operatorname{L}_{\mathbb C} (\gamma) =...
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1 vote

Reference for an easy lemma on homeomorphisms of connected manifolds

I suppose we induct on $n$ (the number of points) and appeal to some version of "Guggenheim's theorem". I started by looking in "Introduction to piecewise linear topology" by ...
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  • 18.5k
1 vote

fills a given polygon with a few types of given primitives

As Sam Nead suggested: Demaine, Erik D., and Martin L. Demaine. "Jigsaw puzzles, edge matching, and polyomino packing: Connections and complexity." Graphs and Combinatorics 23, no. 1 (2007):...
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1 vote

Are two nonsimple closed geodesics in minimal position?

Assuming you are asking about geodesics in a hyperbolic surface, the answer is “yes”. This follows from the “bigon criterion”.
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  • 18.5k
1 vote

Positive vs negative Dehn twist monoids

A general discussion may help clarify these points. There's a lot more to say (e.g. how the positive monoid gives Weinstein fillings, or how negative Dehn twists explicitly give overtwisted disks), ...
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  • 996
1 vote

Fundamental groups of primitive non-algebraic compact Kähler manifolds

I am not sure how one checks primitivity for a compact Kähler manifold, this seems to be too delicate property. Intuitively speaking, very non-algebraic Kähler manifolds have very simple fundamental ...
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