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Results tagged with gt.geometric-topology
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user 6205
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).
12
votes
Accepted
Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersu...
If $[\Sigma]\in H_{n}(M,\mathbb Z)$ is primitive, its image $[\Sigma] \in H_{n}(M,\mathbb Z/2\mathbb Z)$ can always be represented by a non-orientable manifold! Since it is primitive, we may suppose …
- 10.5k
9
votes
Computation on characteristic classes
There is a great theorem proved by Halperlin - Toledo, which says that if you have a triangulated manifold $M$, you can represent combinatorially every Stiefel - Whitney class $w^k(M) \in H^k(M, \math …
- 10.5k
14
votes
In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
One may freely pass from $S^3$ to $\mathbb R^3$ just by adding/removing a point; however $S^3$ is nicer when using 3-dimensional topology techniques, in particular geometrization.
For instance, the co …
- 4,713
4
votes
Accepted
How to distinguish Pretzel links with the same coefficients?
Richard Bedient has proved in 1984 that two Pretzel knots $P$ and $\sigma P$ are equivalent if and only if $\sigma$ is a cyclic permutation, an order reversing permutation, or a composition of both. H …
- 2,821
2
votes
A neighborhood of a 2-disc $D\subset\Bbb R^4$ that tapers off towards the boundary?
Pick a triangulation of $\mathbb R^4$ which contains $D$ as a subcomplex, make two barycentric subdivisions, and then pick the closure of all the simplexes that intersect the interior of $D$. This cou …
- 10.5k
2
votes
Accepted
An equivalence relation on knots similar to concordance
Take a knot, and another copy of the same knot far away. It is then a nice instructive exercise to prove that if they are not unknots they do not cobound annuli.
- 10.5k
42
votes
Accepted
Kirby calculus and local moves
There is a finite set of local moves. For instance, these:
(source: unipi.it)
In the second row, the number of encircled vertical strands is $n\leqslant 3$ on the left and $n\leqslant 2$ on the right …
- 4,713
6
votes
Injectivity of map of fundamental groups from totally geodesic hypersurface
You need $X_0$ to be closed inside $X$, otherwise, the theorem is false (there are simple counterexamples). So let us suppose that $X_0$ is closed.
Inclusions give the induced homomorphisms
$$\pi_1(X_ …
- 563
6
votes
Accepted
Revisiting Gordon-Luecke theorem
I think that the correct theorem should be the following:
If $L$ does not contain any split unknot or any two coaxial components, the image of the map is finite
Following Cameron Gordon, two coaxial …
- 10.5k
15
votes
Accepted
Can we embed a closed manifold into a homotopy equivalent CW complex?
Pick a torus, and add two discs along a meridian and a longitude. You get a 2-complex homotopic to a sphere that does not contain a sphere. This generalises easily to any genus by picking a genus-$g$ …
- 10.5k
9
votes
Accepted
Dehn surgery along primitive knot in 3-dimensional handlebody
Since $c\subset H_g$ intersects an essential disc $D$ in a single point, the boundary of a regular neighbourhood of $D\cup c$ is another disc $D'$, which splits $H_g$ into a solid torus containing $D\ …
- 10.5k
6
votes
Cell structures of Dold manifold and Wu manifold
The 1965 paper Simply connected five-manifolds of Barden contains a simple topological description of the Wu manifold. The Wu manifold decomposes into two copies of the (unique) orientable non-trivial …
- 10.5k
3
votes
Can triangulations (or some related combinatorial structure) distinguish smooth structures o...
It is very hard to construct state-sum invariants that distinguish smooth structures in dimension 4, for this simple but crucial fact that is worth mentioning: if $M$ and $N$ are homeomorphic smooth 4 …
- 10.5k
9
votes
Accepted
Existence of fibered surfaces in arbitrary 4-manifolds?
I suppose that, for a $n$-manifold $M$, containing a "fibered codimension-2 manifold" $N\subset M$ means that $N$ has trivial normal neighbourhood $\nu N = N\times D^2$ and its complement $M \setminus …
- 10.5k
3
votes
Accepted
Geometry of a manifold after Dehn filling, in terms of geometry pre-filling
The general principle is that a generic filling of a geometric manifold belongs to the same geometry of the original manifold. This holds notably in hyperbolic geometry by Thurston's Dehn filling theo …
- 10.5k