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).
4,134
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Graph embedding in 3D grid minimizing edge length
I know that arbitrary graphs can be embedded trivially in $\mathbb{R^3}$ and that planar graphs can be drawn on a plane using Schnyder's grid embedding algorithm after triangulation. And then there is ...
10
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2
answers
1k
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The boundary of a domain whose interior is diffeomorphic to the ball
We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors).
My question is about a very ...
5
votes
1
answer
819
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Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
11
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0
answers
349
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Fox re-imbedding theorem in dimension four
Fox re-imbedding theorem states the following:
A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
1
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1
answer
730
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Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation'
Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ...
2
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1
answer
155
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Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group
Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It ...
9
votes
1
answer
282
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Does the shortest path between two braids pass through string links?
One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...
1
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0
answers
98
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PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
8
votes
1
answer
571
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Virtual fibering conjecture for cusped hyperbolic manifolds
I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the ...
8
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1
answer
697
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Is this knot invariant already treated somewhere in the literature?
Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...
0
votes
1
answer
304
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Torelli group of a punctured elliptic curve
Let $T_{g,n}$ be the Torelli group of a $n$-punctured surface $S=\overline{S}\setminus\{x_1,\ldots,x_n\}$, with $\overline S$ orientable, closed and of genus $g$. By definition, $T_{g,n}$ is the ...
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2
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207
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Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$? [closed]
Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?
4
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0
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Convex subsets of infinite dimensional spaces up homeomorphism
Let $C$ be a convex, infinite-dimensional, non-locally-compact subset of a separable Frechet space.
If $C$ is a closed subset (or more generally, if $C$ is completely metrizable), then it is known ...
3
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2
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Kenji Fukaya's Lecture series at Simons center
In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry.
...
8
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1
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Relation between Milnor ring and middle dimensional homology of hypersurface
I have suspected that the following is well-known:
If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ${\...
3
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2
answers
349
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Handlebody decomposition of a 3-manifold adapted to a link
Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that:
$...
6
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1
answer
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How many distinct homeomorphism classes of lens spaces are there with a fixed p?
This question is about the topological classification of lens spaces. Fix $p$ a positive integer, not necessarily a prime. From Brody, The topological classification of lens spaces, Annals of Math.
(...
15
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3
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Space of embeddings of circle in a surface
Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$.
Question : what is the ...
3
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1
answer
414
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Is any smooth homeomorphism isotopic to a smooth embedding?
Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball).
We call $f$ embedding if it is a homeomorphism on the image and the
derivative $D_xf$ is nonsingular at each point $x\in D^m$ ($\...
5
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0
answers
241
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Ribbon knot presentations
Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and $...
1
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1
answer
299
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Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure
I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...
11
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1
answer
320
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Rank and hyperbolic volume
Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of ...
6
votes
1
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263
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Determining Tightness/Overtwistedness of Contact Structure using Lift of Structure to the Universal Cover
All:
I would appreciate any ideas, refs., etc. on the following:
Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the
contact structure, say $\eta$ on $M^3$ lifts to a ...
10
votes
1
answer
452
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Mapping class group vs automorphism group in cobordism category
Let $3Cob$ be the category whose objects are closed surfaces and whose morphisms are diffeomorphism classes of cobordisms.
By sending a diffeomorphism $\phi$ of a surface $X$ to its associated ...
2
votes
0
answers
26
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compact centralisers in maximal Kac-Moody groups over finite fields
Suppose that I want to choose an element of a maximal Kac-Moody group over a finite field which has a compact centraliser. Is there anything known at all about how I can go about choosing such an ...
5
votes
2
answers
590
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Automorphisms of surfaces, open books and contact structures
Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of $\mathrm{MCG}(S)$, the mapping class group of $S$, i.e., the group of self-diffeomorphisms of $S$ up to isotopy,...
2
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0
answers
207
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Foliation of surface all of whose leaves are circles
I'm having trouble locating a reference for the following basic fact. Let $S$ be a compact orientable surface with boundary. Assume that $\mathcal{F}$ is a foliation of $S$ all of whose leaves are ...
2
votes
2
answers
458
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Origin of the name "Torelli group"
The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface.
The first paper I am aware of that uses the ...
11
votes
1
answer
430
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Construction of the Casson invariant
What is the easiest construction of the Casson invariant? The original construction using representation spaces (as found, for instance, in Akbulut-McCarthy) is very technical since you have to ...
2
votes
1
answer
385
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Module structure of the abelianization of the commutator subgroup
Let $G$ be a (non-abelian) group, and let $G_2$ denote its commutator subgroup. Then the abelianization $G_2^{ab} = H_1(G_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The ...
3
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0
answers
152
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Rigidity vs Super-rigidity of representations (of Kähler/surface groups)
In the literature there are several definitions of "rigidity" (or "super-rigidity") of representations, adapted to the circumastances. I wonder what are the relations between them; I excuse in advance ...
1
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1
answer
225
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Counting edges in embeddable CW-complexes in R^3
Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...
1
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1
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How many types of jigsaw puzzle pieces in n dimensions?
I was partitioning jigsaw puzzle pieces with some friends yesterday and we noticed that there are 6 types of pieces:
All 4 sides have a knobby bit sticking out
1 side has a knobby bit sticking out
2 ...
5
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2
answers
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Realizing homology classes on surfaces with boundary by simple closed curves
Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H_1(\Sigma)$ can be realized by a simple closed curve if there ...
0
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1
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An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable
Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\...
3
votes
1
answer
325
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Going from _Actual_ Open Books to _Abstract_ Open Books
All:
I am looking for a proof of the result that any open book $(B, \pi)$ ; $B$ a fibered link , $\pi$ a map of $M^3-B \rightarrow S^1$ on a 3-manifold $M^3$ , so that $\pi$: $(M^3-B)$ fibers over $S^...
6
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2
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341
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Are there some list of the finite subgroups of the mapping class groups of low genus surfaces?
We already know the bound of the order of the finite subgroups of the $Mod(S_g)$. If we take a further step, to find all the finite subgroups, then what is the result for low genus cases? For example, ...
2
votes
0
answers
50
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What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?
I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry.
One alternative way of formulating it: I am looking for an equivalent of Weaire–...
1
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0
answers
156
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Entangled helical knots
Consider a pair of disjoint, congruent helices $H_1$ and $H_2$
passing through one another in the following sense.
(Caveat lector: This question is not of general interest! It is also long.)
$H_1$ is ...
5
votes
0
answers
167
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are smooth homotopic open embedding of Hilbert manifolds smoothly isotopic?
A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert ...
9
votes
3
answers
760
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What constant ensures hyperbolicity of Dehn surgery?
I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...
2
votes
1
answer
194
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Frohman & Fine's proof about Bianchi groups as HNN extensions (or anyone else's)
Specifically, I am looking for a proof that for squarefree $d\in\mathbb{N}\setminus\{ 1,3\}$, there exists some Kleinian $\Gamma$ and some $\alpha\in$Aut$(PSL(2,\mathbb{Z}))$, such that the Bianchi ...
7
votes
0
answers
427
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Reference Request: Topological h-cobordism theorem in higher dimensions
I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so.
The h-cobordism theorem is true in the topological and in the smooth category in ...
4
votes
2
answers
636
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Action of Mapping Class Group on Arc complex
Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...
0
votes
1
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162
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$S^n$ admit a real polarization $D\subset TS^n$?
When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
7
votes
2
answers
1k
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All mapping space between CW complexes is a CW complex?
Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of $\...
1
vote
1
answer
254
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A problem related to connectivity of analytic functions
Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$.
Question: Is the connectivity of $f(\...
-1
votes
1
answer
298
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Homeomorphism of the punctured sphere which fixes an essential Jordan curve
$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
7
votes
1
answer
384
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High dimensional generalized Poincare hypothesis without the h-cobordism theorem?
The generalized PL Poincare hypothesis states that in dimension $n$ there is a unique PL manifold that has the homotopy-type of $S^n$. It's known to be true in all dimensions except perhaps $n=4$.
...
10
votes
2
answers
353
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Equivariant smoothing of PL structures on $S^3$
Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms?
I am not quite ...