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Results tagged with gt.geometric-topology
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user 280895
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
3
answers
545
views
Contractible set in a manifold
Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F …
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4
votes
1
answer
391
views
3-manifold with boundary containing a projective plane
Let $M$ be a compact $3$-manifold such that no component of $\partial M$ is $S^2$ and one component $F$ of $\partial M$ is the projective plane.
If $i_*:\pi_1(F) \to \pi_1(M)$ is an isomorphism, can w …
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3
votes
1
answer
214
views
One-sided incompressible surface in 3-manifolds
Let $M^3$ be a closed orientable $3$-manifold. If $H_2(M,\mathbb Z)=0$ and $H_2(M, \mathbb Z_2)\ne 0$, can we show that $M$ contains a 1-sided incompressible surface?
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3
votes
2
answers
210
views
$P^2$-irreducibility of a $3$-manifold
A $3$-manifold $M$ is called $P^2$-irreducible if it is irreducible and there is no $2$-sided $P^2$ contained in $M$.
Can we show $M$ is $P^2$-irreducible iff $\pi_2(M)=0$?
Notice that one direction f …
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3
votes
1
answer
115
views
$\pi_1(M^3)$ containing a normal infinite cyclic subgroup
Let $M^3$ be a compact $3$-manifold such that $\pi_1(M)$ contains a normal subgroup isomorphic to $\mathbb Z$.
Can we show either $\pi_1(M)$ is torsion-free or $\pi_1(M)=\mathbb Z \oplus \mathbb Z_2$ …
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3
votes
1
answer
152
views
Surface separating the boundary of a cylinder
Let $M^2$ be a connected closed surface. Suppose there exists an smooth embedding from a connected closed surface $N$ into the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and …
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3
votes
0
answers
60
views
Embedding with vanishing images of homotopy groups
Let $f$ be a locally flat embedding from $S^2 \times \mathbb R^2$ to $S^2 \times \mathbb R^2$ such that $f_*(\pi_k(S^2 \times \mathbb R^2))=0$ for any $k \ge 2$.
Can we find a domain $U$ that contains …
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3
votes
2
answers
189
views
Embedded submanifold in a cylinder
Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates …
- 891
3
votes
1
answer
180
views
Principal circle bundles over punctured $3$-sphere
Let $M$ be $S^3$ with $k$ disjoint open balls $D^3$ removed.
Can we classify all principal circle bundles over $M$ such that the total space is simply-connected?
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2
votes
1
answer
596
views
Classification of disk bundle over surfaces
Are there any reference for the classification of orientable disk bundle over a closed surface? I am particularly interested in the case if the surface is $S^2,RP^2,T^2$ or the Klein bottle.
Many than …
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2
votes
0
answers
118
views
Kahler surface with certain topology
Let $P^2 \mathrel{\tilde \times} \mathbb{R}^2$ be the $\mathbb{Z}_2$-quotient of $S^2 \times \mathbb{R}^2$, where the $\mathbb{Z}_2$ action on $S^2 \times \mathbb{R}^2$ is antipodal on $S^2$ and a ref …
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2
votes
0
answers
132
views
Restriction function as a Morse function
Let $\Sigma$ be a closed surface smoothly embedded in $\mathbb R^3$. For any Morse function $h:\mathbb R^3 \to \mathbb R$, can we isotope $\Sigma$ so that the restriction of $h$ on $\Sigma$ is also a …
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1
vote
1
answer
56
views
Compatibility of two cylindrical regions
Let $M^2,N^2$ be connected closed surfaces. Suppose there exists region $D$ in the interior of $M \times [-2,2]$ such that (a) $D$ is homeomorphic to $N \times [0,1]$; (b) $D$ contains $M \times [-1,1 …
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1
vote
0
answers
160
views
Contractible four-manifold which admits a decomposition
Let $M^4$ be a noncompact, contractible, smooth manifold. Suppose there exists an exhaustion $M=\bigcup_{i\ge 1} U_i$ by open sets such that (1) $\bar U_i \subset U_{i+1}$ and (2) each $U_i$ is homeom …
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1
vote
0
answers
103
views
Extend a circle action on $3$-manifolds
Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.
Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, …
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