Questions tagged [gt.geometric-topology]
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).
4,151
questions
3
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Example of homeomorphism that lifts to real blow up but not C^1?
Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(...
- 41
5
votes
0
answers
274
views
A certain kind of proof of the Hairy Ball Theorem
I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature.
From a given vector field $V_1$ on $S^2$, form another, $V_2$, ...
- 17.5k
8
votes
1
answer
290
views
Topology of a smoothing of an isolated singularity
Let $(X,x)$ be an affine variety with a normal isolated singularity and $Y$ be a smoothing of $X$ (for this we mean $Y$ can be realized as a smooth fibre of a deformation of $X$).
Question. Can we ...
- 81
7
votes
1
answer
224
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If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?
I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 ...
- 1,087
5
votes
0
answers
101
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Standard 2-instantons on the 4-sphere under conformal transformation
It is well-known that there is a standard SU(2) 1-instanton on the 4-sphere and any 1-instanton can be obtained from the standard one by conformal transformation. Is there any explicit (family of) &...
- 653
5
votes
0
answers
153
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The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$
Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal
D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can
$\...
- 1,087
3
votes
1
answer
196
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Algebraic variations of the full knot Floer complex
In Hom's paper (arXiv link), p.20, Section 3.3 ends with
"There are other algebraic modifications one may consider, such as setting $U^n =
0$ or $UV = 0$",
referring to the knot Floer ...
- 141
11
votes
2
answers
1k
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Is there a contractible hyperbolic 3-orbifold of finite volume?
Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...
- 423
5
votes
1
answer
270
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Homology of spherical $3$-manifold group
I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true.
Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
- 179
1
vote
0
answers
83
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about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
10
votes
3
answers
529
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Centre of orbifold fundamental group of torus (Klein bottle) with one cone point
$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely,
$$\...
- 533
0
votes
0
answers
59
views
Obstruction to finding a framing for quotient manifolds
The question is rather open-ended but I hope it is concrete enough.
If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what ...
6
votes
2
answers
384
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How to prove that Lie group framing on S^1 represents the Hopf map in framed cobordism
The Pontryagin-Thom construction gives an isomorphism from the stable homotopy groups of spheres and framed cobordism groups. It seems to be well-established that for dimension 1 (see this question), ...
4
votes
0
answers
99
views
Mostow-Palais equivariant embedding for manifolds with corners
Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
- 373
6
votes
0
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96
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Classification of contractible open n-manifolds which embed in a compact n-manifold
Does there exist a classification of contractible open $n$-manifolds ($n\geq 3$) which embed in a compact $n$-manifold? More general, does there exist a classification of contractible open $n$-...
- 1,936
17
votes
1
answer
475
views
Topology of the space of embedded genus $g$ surfaces in $S^3$
Consider the space of smoothly embedded genus $g$ surfaces in the 3-sphere under the $C^\infty$ topology:
$$\mathcal E_g:=\operatorname{Emb}(\Sigma_g,S^3)/{\operatorname{Diff}(\Sigma_g)}$$
where $\...
- 513
5
votes
3
answers
227
views
Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$
Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
- 2,483
3
votes
1
answer
136
views
Existence of covering isomorphism
Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p_1, p_2: C \rightarrow D$ are given and have the same degree. Is there always an ...
- 63
5
votes
1
answer
284
views
Stable torus that is not a torus [duplicate]
Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?
- 43.7k
6
votes
0
answers
338
views
Arithmetic Teichmüller curves, first eigenvalue of the Laplacian, McMullen's expander conjecture
$\DeclareMathOperator\SL{SL}$By a result due to Ellenberg and McReynolds, any finite index subgroup $\Gamma$ of $\Gamma(2) \subset \SL\left(2,\mathbb{Z}\right)$ is the Veech group of an arithmetic ...
- 11
3
votes
0
answers
342
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Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 321
5
votes
1
answer
242
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Question about and good reference for Kahn and Markovic result
As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:
Let $M$ be a compact hyperbolic $3$...
- 829
4
votes
2
answers
116
views
Action of noncentral mapping classes on curves or arcs on a surface
$\DeclareMathOperator\MCG{MCG}$Let $\Sigma$ be a compact oriented surface, with empty or connected boundary. Let $\mathcal{O}$ the space of orbits of nontrivial simple closed curves on $\Sigma$ under $...
- 492
2
votes
1
answer
196
views
Knot concordance, hyperbolicity and amphichirality
Let $K_0$ and $K_1$ be two knots in $S^3$. We say $K_0$ and $K_1$ are concordant if there exists a smoothly embedded annulus $A \subset S^3 \times [0,1] $ such that $\partial A = -(K_0) \cup K_1$.
...
- 606
9
votes
1
answer
488
views
Are any embeddings $[0,1]\to\mathbb{R}^3$ topologically equivalent?
Suppose we are given embeddings $f_1,f_2:[0,1]\to\mathbb R^3$.
Does there exist a homeomorphism $g:\mathbb R^3\to\mathbb R^3$ such that $g\circ f_1=f_2$?
This question seems to be classical eighty ...
- 965
1
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0
answers
226
views
Presentation complexes with same homology and different fundamental groups
If we start with a perfect group $G$ of deficiency zero then there is a presentation $P$ of $G$ such that the number of relations and the number of generators for $P$ are the same. For such $P$, the ...
- 179
4
votes
0
answers
95
views
Presentation of handlebody mapping class group
I know some 'nice' infinite presentations of the mapping class group of a surface, such as Gervais' and Luo's. By 'nice' I mean that generators and relations belong to a small number of families.
Is ...
- 361
6
votes
1
answer
378
views
Two surfaces in a 4-manifold whose algebraic intersection number is zero
Suppose $X$ is a smooth closed oriented 4-manifold, and $\Sigma_1,\Sigma_2$ are smoothly embedded compact oriented surfaces in $X$. Suppose they intersect transversally at two points with different ...
- 475
1
vote
0
answers
137
views
Simplicial sets and oriented simplicial complexes
$\DeclareMathOperator\Sing{Sing}$I'm writing a paper about simplicial sets and how they may “replace” simplicial complexes in some known results. To do this, we need to check that they induces the ...
- 11
11
votes
0
answers
210
views
On an Artin (?) subgroup of braid groups
While working on something apparently unrelated I encountered a "braid-like" group, which is a relatively geometric subgroup of a braid group and seems to be itself an Artin group. It seems ...
- 40.2k
4
votes
1
answer
205
views
Stallings' fibration theorem - Explicit description
Stallings' celebrated Fibration Theorem states that if a closed irreducible $3$-manifold $M$ admits a short exact sequence
\begin{equation}
1 \to N \to \pi_1(M) \to \mathbb{Z} \to 1,
\end{equation}
...
- 143
4
votes
1
answer
226
views
Homology classes in connected sum of $\Bbb CP^2$'s that can be represented by smoothly embedded spheres
Let $h=[\Bbb CP^1]\in H_2(\Bbb CP^2;\Bbb Z)$. By a theorem of Kronheimer and Mrowka (Theorem 1 of this paper: https://people.math.harvard.edu/~kronheim/thomconj.pdf), a class $nh \in H_2(\Bbb CP^2;\...
- 475
3
votes
0
answers
96
views
Kauffman bracket for Abelian anyons
The Kauffman bracket, defined here, assigns a polynomial in $A$ to any knot. (For concreteness consider the Kauffman bracket normalized so that the unknot is assigned $-(A^{-2}+A^2)$.) For certain ...
- 31
6
votes
1
answer
184
views
Bigon criterion in dimension 3?
The bigon criterion for surfaces says that if two simple closed curves $\alpha$ and $\beta$ embedded on a surface $\Sigma$ intersect in points $\{p_1,\dotsc,p_n\}$ and $\alpha$ and $\beta$ can be ...
- 1,065
10
votes
0
answers
189
views
"Homotopy homomorphisms" of homeomorphisms of Euclidean space
For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...
- 7,933
4
votes
0
answers
107
views
topological and $\mathscr{C}^{\infty}$-circle bundles
Let $M$ be a closed $\mathscr{C}^{\infty}$-manifold. Suppose that the underlying topological space of $M$ has a topological circle bundle structure $S^1\hookrightarrow M\to B$. Does $M$ admit a $\...
- 113
3
votes
1
answer
142
views
Spaces satisfying a strong Cartan-Hadamard theorem
Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a local-isometry where $H$ is either a Hilbert space or a Euclidean space?
...
- 310
2
votes
1
answer
187
views
Estimating the volume of a convex shape in higher dimensions based only on normal sections
We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ ...
- 1,677
3
votes
0
answers
411
views
Show that continuous maps between smooth manifolds can be approximated by smooth maps WITHOUT using Whitney's embedding theorem
As it is well-known (and for example this question shows) each continuous map between smooth manifolds is homotopically equivalent to a smooth map that can be constructed using the Whitney embedding ...
- 1,037
3
votes
1
answer
132
views
Is there an analogy of Austin-Braam approach to Bott-Morse type Hamiltonian Floer homology?
Austin-Braam approach uses the multicomplexes of de Rham complex on critical submanifolds to describe Bott-Morse theory.
For more details, see the follows:
https://link.springer.com/chapter/10.1007/...
- 265
2
votes
0
answers
144
views
"Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\CC{C}$In the article "Hyperbolic rigidity of higher rank lattices", Thomas Haettel has proved the following theorem: Let $\Gamma$ be a ...
- 187
1
vote
1
answer
241
views
Name for extension of the symplectic group
Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\...
- 925
5
votes
1
answer
148
views
Nonexistence of sphere with one conical point [reference request]
It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven ...
- 458
5
votes
1
answer
193
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Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?
In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3_{\Bbb Z}\to \Bbb Z$, where $\Theta^3_{\Bbb Z}$ is ...
- 475
8
votes
1
answer
622
views
On trivial mapping class group of 3-manifolds
What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
- 3,571
3
votes
0
answers
190
views
Is there such an isotopy for every homology sphere?
Let $n \geq 3$, and $\Sigma^{n-1} \subset \mathbf{S}^n$ be a smoothly and properly embedded, orientable, and connected submanifold of the sphere. This divides the sphere into two open sets, $U_-$ and $...
- 4,968
7
votes
1
answer
419
views
Small knots becoming isotopic after connect sum
I am interested in the following situation: I have two codimension-2 knots $K_1$ and $K_2$ in $S^n$ and they are not isotopic. Furthermore, $K_1$ is not isotopic to the mirror image of $K_2$ and ...
- 1,065
3
votes
0
answers
119
views
Is there a framed nullbordism of $T^4$ with an action of $T^4$ that extends the self-action?
Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. ...
- 1,982
3
votes
0
answers
142
views
A question about index of Dirac operator
Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...
- 533
3
votes
1
answer
200
views
Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?
It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed ...
- 33