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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).
23
votes
2
answers
1k
views
Representing elements of $\pi_2(M)$ by embedded spheres in 3-manifolds
I am sorry that this question is probably too basic - I could not seem to find the answer though. I know the following - let $S$ be a closed orientable surface, an element of $H_1(S;\mathbb{Z})$ is r …
- 5,349
19
votes
2
answers
700
views
Behavior of genus function on a 4-manifold for sums
Let $X$ be a smooth compact 4-manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \mathb …
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15
votes
1
answer
408
views
Equivalence of surjections from a surface group to a free group
Let $g \geq 2$. Let $S = \langle a_1,b_2,...,a_g,b_g | [a_1,b_1] \cdots [a_g,b_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given tw …
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13
votes
0
answers
177
views
Is there a Handle Approximation theorem?
The cellular approximation theorem states that given a continuous map between two CW complexes $f : X \to Y$, then $f$ is homotopic to a cellular map - that is some map $f'$ with $f'(X_n) \subset Y_n$ …
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12
votes
2
answers
487
views
Minimal area of Seifert surfaces
Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\math …
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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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10
votes
1
answer
767
views
Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?
Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and …
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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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9
votes
1
answer
307
views
Genus 2 3-manifolds bounding only $X^4$ with $b_2(X^4)$ big?
The genus of a closed orientable 3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$. Every such 3-manifold bounds a compact 4-manifold. Let $I(M)$ denote the minimu …
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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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9
votes
1
answer
417
views
Non-isotopic homology spheres in $S^4$ with equal complements?
Are there two diffeomorphic smoothly embedded homology 3-spheres $M_1^3, M_2^3 \subset S^4$ that have diffeomorphic complements but such that $M_1$ and $M_2$ are not isotopic? I would be interested in …
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9
votes
1
answer
765
views
Intuition for torsion of a chain complex and application to lens spaces
I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that classif …
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8
votes
0
answers
171
views
Local formula for the signature of $4k$-manifold
In "The Euler characteristic is the unique locally determined homotopy invariant of finite complexes" on page 61 in the penultimate paragraph, Levitt mentions that if one restricts to compact PL $4k$- …
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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 …
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8
votes
2
answers
321
views
Manifolds with trivial mapping class group and large $H^1$?
Are there smooth closed manifolds $M^n$ in every dimension $n \geq 3$ with trivial mapping class groups and with $H^1(M^n;\mathbb{Z}/2\mathbb{Z})$ arbitrarily large?
I am under the impression that "ge …
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