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Results tagged with gt.geometric-topology
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user 39082
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).
4
votes
Accepted
laminations and branched surfaces
Tao Li https://www2.bc.edu/~taoli/lbs.pdf constructs an essential lamination for each branched surface satisfying the following conditions:
(1) Its horizontal boundary is incompressible; (2) there is …
- 10.4k
2
votes
Accepted
Are all transversely oriented, transversely measured foliations given by closed forms?
The claim is true and you find it on page 319 of Farb-Margalit's book.
The point is that the transition maps between foliation charts (with the leaves as y-level sets) are of the form $(x,y)\to (f(x, …
- 10.4k
0
votes
Cyclic groups acting on balls, and interior fixed points
Assume all fixed points are on the boundary. For each fixed point you can find an invariant neighborhood. (You can assume that the $Z/nZ$-action fixes some Riemannian metric. Then any $\epsilon$-ball …
- 10.4k
0
votes
Circle Bundles of surfaces
Flat orientable circle bundles (topological, not necessarily linear) are classified by homeomorphisms of the fundamental group to $homeo^+(S^1)$, and these are in a weak sense (up to topological semic …
- 10.4k
4
votes
Attaching a thickened annulus between two 3-manifold
Let me try to give a more elementary argument.
Assume there is a compression disk. One can assume it is transversal to $A\times\left\{\frac{1}{2}\right\}$, thus it intersects $A\times\left\{\frac{1}{ …
- 10.4k
3
votes
Does Dehn filling always decrease Gromov norm?
In fact, for manifolds with torus boundary you have $$\parallel M,\partial M\parallel_0=\parallel M,\partial M\parallel$$ so that the two inequalities are plainly equivalent.
(More generally, this eq …
- 10.4k
1
vote
Does Dehn filling always decrease Gromov norm?
There is also an explicit argument which shows
$$\parallel M_1\cup_{A}M_2\parallel \le \parallel M_1,\partial M_1\parallel + \parallel M_2,\partial M_2\parallel$$
whenever $A=\partial M_1=\partial M_2 …
- 10.4k
15
votes
Accepted
Homology sphere with $\mathbb{R}^3$ as the universal cover
In a sense, most $3$-manifolds have universal cover $R^3$. In particular, this is the case for hyperbolic $3$-manifolds. And there do exist integer homology spheres which are hyperbolic. Two explicit …
- 10.4k
1
vote
Accepted
Non-positive sectional curvature in 3-dimensional manifold
Edited.
From Theorem 5.3 in https://homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf a circle bundle over a hyperbolic surface has a geometry locally modeled on either $H^2\times R$ or …
- 10.4k
9
votes
Accepted
3-dimensional h-cobordisms
For $M_0=M_1=S^2$, this follows from the 3-dimensional Poincaré conjecture: glueing in 3-balls, you get a simply connected $3$-manifold, that has to be $S^3$, so $W=S^2\times\left[0,1\right]$.
For sur …
- 10.4k
2
votes
Accepted
Construct a homeomorphism between two surfaces
A closed surface of genus g can be cut along 2g closed curves (all at one base point) to obtain a 4g-gon. Do this for both surfaces. Then choose a homeomorphism between the 4g-gons which matches the c …
- 10.4k
10
votes
1
answer
273
views
Complements of unknotted tori (higher dimensions)
It is well-known that an unknotted 2-torus in $S^3$ provides the standard Heegaard splitting, in particular its complement consists of two solid tori.
It is also known that an unknotted 3-torus in $S …
- 10.4k
5
votes
Is there a smooth manifold which admits only rigid metrics?
The survey paper "Do manifolds have little symmetry?" by Volker Puppe lists several known results about manifolds with no nontrivial finite group action. The ArXiv link is http://arxiv.org/pdf/math/06 …
- 10.4k
4
votes
What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$
$P\times S^1$ has an obvious fibration by circles and as long as your Dehn filling does not send a meridian to the fiber of that fibration, the Dehn filled manifold will again be a Seifert fibration. …
- 10.4k
3
votes
Accepted
Handle body of 3-manifold with boundary
3-dimensional handlebodies are obtained by gluing 1-handles to a 3-ball. It is true, that every compact, orientable 3-manifold has a handle decomposition, but this needs not just 1-handles but also 2- …
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