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Results tagged with gt.geometric-topology
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user 23571
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).
28
votes
Accepted
Unique smooth structure on 3-manifolds
An alternative to Moise's paper for the existence and uniqueness of piecewise linear (PL) structures on topological 3-manifolds is the paper "The triangulation of 3-manifolds" by A.J.S. Hamilton in Qu …
- 20k
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 t …
- 20k
51
votes
Accepted
Triangulating surfaces
[Three years later …]
All the published proofs of triangulability of surfaces that I am aware of use the Schoenflies theorem, which is not exactly an easy thing to prove. There is however another line …
- 20k
2
votes
Accepted
Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures
The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk …
- 20k
6
votes
Accepted
generators for the handlebody group of genus two
As stated in Ian Agol's answer, the mapping class group of a handlebody $B$ maps onto $Out(\pi_1B)$. This is easy to show by lifting known generators for the automorphism group of a free group. Twist …
- 20k
12
votes
Accepted
Does there exist a Haken manifold where all its incompressible surfaces are non-separating?
There exist closed orientable hyperbolic 3-manifolds that are surface bundles such that the fiber is the only incompressible surface in the manifold (up to isotopy). Such manifolds can be obtained by …
- 20k
10
votes
Accepted
Handle decompositions using only 1-handles
The second statement ought to be in the literature somewhere but I don't know a reference so I'll give an argument.
The result can be rephrased in terms of graphs. Let $S$ be a compact connected sur …
- 20k
14
votes
Accepted
Mapping class group of certain 3-manifolds
Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to fiber …
- 20k
13
votes
Accepted
Proof of the stable homeomorphism conjecture
For your first question, yes Kirby did prove the n-dimensional annulus conjecture AC$_n$ for $n>4$ in the 1969 Annals paper that you cite. The only reason this might not be clear from reading the pape …
- 20k
7
votes
Accepted
positions of regular cubes in Euclidean space with all its vertices without distinction
If the polyhedron $P$ in ${\mathbb R}^3$ has orientation-reversing symmetries then the orbit space $O(3)/Sym({P})$ is equal to $SO(3)/G$ for $G=G{(P})$ the orientation-preserving elements of $Sym{(P}) …
- 20k
16
votes
Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
Every topological manifold has a handlebody structure except in dimension 4, where a 4-manifold has a handlebody structure if and only if it is smoothable. This is a theorem on page 136 of Freedman a …
- 20k
2
votes
Accepted
Maps of balls with fixed value along boundary
This set of homotopy classes is in bijective correspondence with $\pi_3(M)$. More generally, let $[B^k,X;f]$ be the set of homotopy classes of maps $B^k\to X$ that restrict to a given $f:S^{k-1}\to X$ …
- 20k
4
votes
Accepted
Action of Mapping Class Group on Arc complex
The quotient complex was studied in a paper by Penner, "The structure and singularities of quotient arc complexes", Journal of Topology 1 (2008), 527-550. An earlier version of the paper is available …
- 20k
15
votes
Accepted
Mapping class groups of small Seifert-fibred 3-manifolds
The determination of mapping class groups of small Seifert manifolds was completed by M. Boileau and J.-P. Otal in a paper in Invent. Math. 106 (1991), 85-107. They give references for cases previous …
- 20k
10
votes
One question on cup product and torsion elements
Here's an example that's a 2-dimensional CW complex. Start with a 0-cell, then attach three 1-cells labeled $a$, $b$, $c$ to get a wedge of three circles, then attach a 2-cell via the word $aba^{-1}b^ …
- 20k