15
votes
dichotomy in hyperbolic groups
No, there is no such dichotomy. If $G$ is an infinite group with Property (T) and $H$ is any non-trivial group, then $G*H$ has neither of the two properties. This is because groups with Property (T) ...
- 2,386
12
votes
4-manifold $M$ with intersection form of Leech lattice
If you're assuming $M$ is simply-connected, then it would be spin (since the Leech lattice is even). So a smooth manifold would violate Rokhlin's theorem. In the topological case (still assuming ...
- 18.7k
9
votes
Accepted
Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?
I think the right generality to restrict to is the following:
Let $n$ be a positive integer, and let $X$ and $Y$ be $n$-dimensional oriented simplicial complexes, with the following properties:
Every ...
- 7,279
6
votes
0-surgery on a fibered hyperbolic ribbon knot
I had a quick look at Maggie Miller's thesis. It appears that the pretzel knots $(\pm 2, n, -n)$ where $n$ is odd are fibered ribbon knots (and they are also hyperbolic). Moreover, it appears that the ...
- 64.2k
5
votes
Finding a hyperbolic metric with geodesic boundary on a given Riemann surface
See Theorem 1(b) in the following paper; it is enough that the boundary is smooth.
Osgood, B.; Phillips, R.; Sarnak, P. Extremals of determinants of Laplacians. J. Funct. Anal. 80 (1988), no. 1, 148--...
- 1,338
5
votes
Knot theory in handlebodies of arbitrary genus
You example of graphs in the plane "becoming embedded via local operations" is not modelled by handle attachment. (For example, if you immerse $K_5$ in a disk $D^2$ and then attach one-...
- 22.8k
5
votes
Knot theory in handlebodies of arbitrary genus
If you mean adding 1 and 2 handles to the boundary of the ball, that will never change the (non)triviality of the knot in the 3-ball. On the other hand, perhaps the notion of tunnel number is what you ...
- 564
5
votes
Does every triangulable manifold have a vertex-transitive triangulation?
This is to supplement Ian's answer and get examples in all dimensions $\ge 3$.
Let $M={\mathbb H}^n/\Gamma$ be a compact hyperbolic $n$-manifold; suppose that $f\in Homeo(M)$ is a homeomorphism of ...
- 30.6k
5
votes
Does every triangulable manifold have a vertex-transitive triangulation?
There exists many closed connected hyperbolic 3-manifolds $M$ with trivial symmetry group, and hence trivial mapping class group. $M$ cannot be homeomorphic to a simplicial complex $\tau$ which admits ...
- 64.2k
5
votes
Accepted
Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?
$\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}\def\PP{\mathbb{P}}$I think the answer is no. Topology isn't my strength, so check this argument.
Let $\infty$ be the point $[0:0:1]$ in $\CC \PP^2$ and put $U = ...
- 145k
4
votes
Accepted
Finding a hyperbolic metric with geodesic boundary on a given Riemann surface
A good reference is
W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1).
The idea is that you construct the double: it is the result of gluing of your ...
1
vote
Accepted
Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$
The equations $\sum_{i=1}^6 y_i^k = 0$ for $k=1,2,4,7$ do not cut out a curve because the $k=1,2,4$ equations imply the one for $k=7$. (The 7th power sum is a polynomial in the power sums of degree 1 ...
- 75.2k
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