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Results tagged with gt.geometric-topology
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user 99414
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).
8
votes
1
answer
304
views
Obstructions to realizing a balanced presentation as a 3-manifold group
I am thinking of 3-manifolds as arising from Heegaard splittings which I am thinking about in terms of Heegaard diagrams. I know that 3-manifold groups are rather special in the class of all finitely …
- 5,349
5
votes
2
answers
804
views
Open book decompositions of $T^3$
Please pardon my ignorance on the subject of open books, I'm a noob. I would like to know some explicit descriptions of open book decompositions of the three torus $T^3$. Are there examples with con …
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6
votes
1
answer
537
views
smooth homotopy 4-balls with sphere boundary in dimension 4
What follows is, as far as I can tell, totally standard folklore. I have one particular point of confusion, other than that, I wanted to confirm that I am uttering the incantations correctly.
The …
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7
votes
1
answer
305
views
Homotopy in $X$ and homology in $X \times I$
Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f_1,f_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f_1$ and $f_ …
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9
votes
2
answers
492
views
Existence of fibered surfaces in arbitrary 4-manifolds?
It is apparently a result of F. González-Acuña that all closed orientable 3-manifolds contain a fibered knot. (I am not sure exactly where to find a published proof of this result and as an aside I w …
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5
votes
1
answer
310
views
Is there a generalized Property P - what can we say about framed link descriptions of $S^3$?
A knot $K$ is said to have Property P if every nontrivial Dehn surgery on $K$ yields a 3-manifold that is not simply connected. It is known that every knot except the unknot has Property P. I am won …
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5
votes
0
answers
112
views
When do unknots in $S^n$ bound unique balls?
I recently heard the it is an open question whether or not an unknotted $S^2$ in $S^4$ bounds a unique $B^3$ in $S^4$, where by unique, I mean up to isotopy rel boundary in $S^4$. The rel boundary co …
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10
votes
0
answers
176
views
Embedding 2-complexes null homotopically into 2-complexes
Whitehead's conjecture states that if $L$ is an aspherical 2-complex and $K$ is a subcomplex of $L$, then $K$ is also aspherical. It is known by work of Howie and Luft that if the Whitehead conjectur …
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4
votes
1
answer
194
views
Minimal genus of characteristic surfaces?
Let $X^4$ be a simply-connected closed smooth 4-manifold. Then every element $x \in H_2(X; \mathbb{Z})$ can be represented by an embedded orientable surface and the minimal genus of such a surface is …
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3
votes
0
answers
59
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Nonuniqueness of Heegaard surfaces for submanifolds of $S^3$
Let $M^3$ be some compact submanifold of $S^3$ with connected boundary. I am interested in the failure of the analog of Waldhausen's theorem for $M^3$ - namely, I would like examples of such $M$ toge …
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2
votes
1
answer
125
views
Images of boundary surfaces in 3-manifold groups
Let $M$ be a compact connected 3-manifold and let $S$ be a closed connected surface in $\partial M$. Let $G$ be the image of the map $\pi_1(S) \to \pi_1(M)$ induced by inclusion. I was reading the f …
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8
votes
2
answers
376
views
What does the matrix of a mapping class tell you about the 3-manifold?
Let $H$ be a handlebody with $\Sigma = \partial H$. Given an automorphism $f : \Sigma \to \Sigma$ we can glue to obtain a closed 3-manifold $M = H \cup_f H$ and in fact all such 3-manifolds are obtai …
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4
votes
1
answer
450
views
Representing homology classes in a Heegaard diagram
Given a Heegaard diagram $(\Sigma, \alpha, \beta)$ we obtain a compact 3-manifold $M$ together with a handle decomposition where the $\alpha$ curves are the belt spheres of the 1-handles and the $\bet …
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11
votes
1
answer
397
views
Existence of normal microbundles
In the same paper where Milnor introduced the concept of microbundles, he gave the following definition. $M$ has a microbundle neighborhood in $N$ if there is a neighborhood $U$ of $M$ in $N$ and a r …
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7
votes
1
answer
587
views
Difficulty with "On fibering certain 3-manifolds" by Stallings
I am reading the paper "On fibering certain 3-manifolds" by John Stallings and I was hoping someone could help me through a certain detail. In particular, I am confused at the very end of the proof o …
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