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).
4,155
questions
2
votes
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233
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Length of a simple closed curve under Pseudo-Anosov maps
Let $S$ be a fixed hyperbolic surface with genus $g$ and $n$ punctures. Given any pseudo-Anosov map $f$ on $S$ (with stretch factor $\lambda$) with stable and unstable measured foliations $\mu^s$ and $...
- 1,703
4
votes
2
answers
687
views
Are there Kirby diagrams with 3-handles?
Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\...
- 5,545
5
votes
1
answer
211
views
Pull-back of knots in branched covers and the Alexander polynomial
Given a knot $K \subset S^3$ one can form its double branched cover $\Sigma_2(K)$ and consider the pull-back knot $\widetilde{K} \subset \Sigma_2(K)$ of $K$ to $\Sigma_2(K)$ (the locus fixed by the ...
- 606
26
votes
4
answers
895
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Why do some uniform polyhedra have a "conjugate" partner?
While browsing through a list of uniform polynohedra, I noticed that the square of the circumradius $R_m$ of the small snub icosicosidodecahedron ($U_{32}$) with unit edge lengths is,
$$R_{32}^2 =\...
- 12.1k
3
votes
0
answers
230
views
Pairs of non-isometric subsurfaces of a hyperbolic 3-manifold, with the same genus
When I say manifold below, I mean a complete orientable finite-volume hyperbolic $3$-manifold and when I say subsurface, I mean immersed closed totally geodesic subsurface.
Whenever a manifold has a ...
- 2,438
16
votes
3
answers
892
views
Relationships between homology maps of cobordant manifolds
Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$.
Does anybody know of any nice examples of general relationships between the images ...
- 161
8
votes
0
answers
197
views
Spin structures and bitangents
I came across an interesting remark in Atiyah's classic paper, Riemann surfaces and spin structures. He computes that there are 28 spin structures on a genus 3 surface with Arf invariant 1, and then ...
- 19.2k
7
votes
3
answers
722
views
Bounded cohomology motivation
May I ask what is the basic motivation behind studying bounded cohomology?
Is there a simple reason why bounded cohomology is more interesting / useful than usual cohomology?
Also, is bounded ...
- 1,219
4
votes
1
answer
209
views
Trace field of a hyperbolic $3$-manifold with a totally geodesic subsurface
Let $X$
be a complete finite-volume orientable hyperbolic $3$-manifold,
and let $\Gamma$
be a Kleinian representation of $\pi_1(X)$.
Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big)...
- 2,438
22
votes
1
answer
699
views
What is the cohomological dimension of the commutator subgroup of the pure braid group?
I'm interested in computing the cohomological dimension of the commutator subgroup $[P_n,P_n]$ of the pure braid group $P_n$. I wasn't able to find a reference in the literature.
Because $[P_n,P_n]$ ...
6
votes
3
answers
518
views
Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
- 3,571
9
votes
2
answers
664
views
Examples of automorphic forms over $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$
If I understood my automorphic forms correctly, at least cusp forms can be thought of as elements of $L^2(G/\Gamma)$ for a $G = \text{SL}_2(\mathbb{R})$ and $\Gamma = \text{SL}_2(\mathbb{Z})$ or a ...
- 22.6k
6
votes
2
answers
1k
views
Are irreducible subgroups Zariski-dense?
A subgroup $H$ of an algebraic group $G$ is said to be Zariski-dense if its Zariski closure is all of $G$ (or alternatively, if every polynomial which vanishes on all elements of $H$ vanishes ...
- 927
3
votes
0
answers
73
views
Two questions regarding flat fibre bundles and the corresponding group action on the fibre
Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...
- 394
6
votes
1
answer
222
views
Asphericity of 2-complexes
Is it decidable whether a finite group presentation is diagrammatically aspherical (that is there is no reduced spherical diagram over this presentation)? Probably - not, but I cannot find a reference....
user6976
6
votes
0
answers
290
views
Lower central series and Euler characteristics of aspherical $2$--complexes
Stallings proves the following (remarkable theorem!) using the lower central series. (Homology and central series of groups, p171.)
Suppose $G$ is a group, $K$ a field, with $H_2(G,K)=0$. If $x_1,\...
- 396
11
votes
3
answers
518
views
Quasi-isometric groups without common virtual geometric model
Are there known examples of finitely generated groups $G$ and $H$ that are quasi-isometric but do not admit finite-index subgroups $G'<G$ and $H'<H$ such that both $G'$ and $H'$ admit proper and ...
- 2,050
3
votes
1
answer
273
views
Symmetry of functions on $S^2$
Let $f$ be a continuous function on $S^2$ and suppose there exists a constant $C>0$ such that for every $\mathcal{R} \in SO(3)$ the area of every connected component of $\{f(x)\geq f(\mathcal{R}x)\}...
4
votes
0
answers
184
views
Right-veering periodic automorphisms of surfaces that are not composition of right-handed Dehn twists
Basically the title of the question. For the sake of completeness I state an introduction to the question.
In "Right-veering diffeomorphisms of compact surfaces with boundary I" and II, the authors ...
- 1,379
2
votes
0
answers
204
views
Sylvester-Gallai-type theorem for quadratic polynomials
Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ ...
- 2,683
9
votes
2
answers
363
views
Are pseudo-Anosov foliations dense?
A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of ...
- 2,370
8
votes
1
answer
210
views
Well definedness of square roots of separating Dehn Twists
Let $c_1,c_2,c_3,c_4,c_5$ be a five chain of circles on a genus 2 surface (i.e $i(c_k,c_{k+1})=1$ and zero otherwise). Then $(T_{c_1} T_{c_2})^6 = (T_{c_4} T_{c_5})^6 = T_c$ where $c$ is a separating ...
- 365
5
votes
1
answer
197
views
Cup product on flat fiber bundles vs cup product on the corresponding Serre spectral sequence
Let $F \rightarrow E \rightarrow B$ be a flat fiber bundle, $E, F, B$ closed manifolds. Consider $H^*(E, \mathbb{Q})$ and the corresponding Serre spectral sequence with isomorphism $$(*) \ \ \ H^n(E;\...
- 394
1
vote
1
answer
138
views
Confusion about locally cone-like spaces
Definition: A filtered space $X$ of formal dimension $n$ is locally cone-like if for all $i$, $0 \le i \le n$, and for each $x \in X^i - X^{i-1} = X_i$ there is an open neighborhood $U$ of $x$ in $X_i$...
- 35
1
vote
1
answer
147
views
Conull subspace containing orbit of an (ergodically acting) group
I should probably start with a warning that this is my first post in this board and that I am sorry, if it is not up to standards. It would be great, if you could let me know how to improve the post.
...
- 13
9
votes
1
answer
713
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 ...
- 5,319
20
votes
3
answers
3k
views
What is the "serious" name for the topograph (for a quadratic form)
One way to study (mixed signature) quadratic forms in two variables is to study the topograph. Looks like the signature doesn't matter: here is (1,1) and (1-,1).
The name is derived from τοποσ (...
- 22.6k
2
votes
1
answer
335
views
Goeritz matrix and link coloring
For a link $L$ and a prime $p$, $L$ has a $p$-coloring iff $p$ divides the $\operatorname{gcd}$ of the invariant factors of the Goeritz matrix of $L$.
Do you know the elementary proof of this facts?
11
votes
0
answers
579
views
What is known about mapping class groups of 4-manifolds?
I am mostly interested in the case when you have a smooth degree $d$ algebraic surface $X$ over $\mathbb C$ and we can define three distinct groups: $\pi_0(\mathrm{Diff}^+(X))$, $\pi_0(\mathrm{Homeo}^+...
- 765
6
votes
0
answers
158
views
Reference to the theorem about linear bundles
The following fact looks like it is well-known. I can prove it myself, but I would like to know of a reference which has a proof?
Let $M$ be an $n$-manifold, $C\subset M$ a codimension-1 closed ...
- 965
1
vote
1
answer
212
views
What’s the relation between Lehmer’s Conjecture and Systole [closed]
What’s the relation between Lehmer's Conjecture and Systole?
Lehmer’s conjecture says: There exists $m>1$ such that $M(p)\geq m$ for all noncyclotomic $P$.
Systole is a closed geodesic of the ...
- 97
1
vote
1
answer
128
views
Uncountability of admissible topological stratifications
I'm not sure if this is trivial. Can someone please provide an example of a space which admits uncountably many topological stratifications? How about up to homeomorphism?
- 35
6
votes
2
answers
312
views
Can cobordisms of 3 or 4 manifolds be visualized by moves on kirby diagrams?
I'm mostly interested in the 4d case so I'll state the question in that form. Basically, it comes down to two parts:
1) What simple moves can be performed to a kirby diagram of a 4-manifold that ...
- 619
6
votes
1
answer
176
views
Are the pure genus zero mapping class groups residually torsion-free nilpotent?
Are the pure genus zero mapping class groups residually torsion-free nilpotent?
They are
-a quotient of the pure braid groups (which are residually torsion-free nilpotent).
-torsion-free.
- 1,424
7
votes
2
answers
717
views
Does there exist a Haken manifold where all its incompressible surfaces are non-separating?
Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A ...
- 3,571
6
votes
1
answer
253
views
Immersions of manifolds with boundary (regular homotopy classes, h-principle)
Have regular homotopy classes of immersions of manifolds with boundary been studied? i.e. immersions $(M, \partial M) \looparrowright (N, \partial N)$. The references I've looked at from this question ...
- 176
4
votes
1
answer
154
views
Twisting equivalent links and the isotopy type of the resulting links
Take any link $L_1$ with an unknotted component
$K$, cut along the disk bounded by K (which usually intersects some of the
other components of $L_1$ transversally ), twist n times, and reglue. Let us ...
- 205
2
votes
1
answer
110
views
Maximum genus of an abstract "cycle complex"
Let us define an abstract "cycle complex" as the following combinatorial object: it is $(V, C)$, where $V$ is a set of $n$ nodes, $C$ is a set of $c$ cyclically ordered subsets of $V$, each ...
- 1,379
3
votes
0
answers
141
views
Seifert-Fibered 3-Manifolds and Rotation Numbers
I was trying to understand how the "ziggurats" come about in the paper by Calegari and Walker.
Motivating Question Given a free group F, and an element w of F, and given
values of the rotation ...
- 22.6k
3
votes
0
answers
160
views
A perfectly round and optimal version of Antoine’s necklace
Suppose we wish to construct a round "necklace" whose links are congruent solid tori, each similar to the toroidal tube containing the necklace (see drawing below). The tube, hence each of the links ...
- 7,256
8
votes
2
answers
553
views
Jordan curves in $\mathbb R^n$ and inscribed equilateral triangles
Inscribed square problem wants that we know "Does every Jordan curve admit an inscribed square?"
From my amateur viewpoint it seems that the concept of Jordan curve can be straightforwardly ...
user114642
30
votes
2
answers
2k
views
Does there exist any non-contractible manifold with fixed point property?
Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
- 3,571
3
votes
0
answers
93
views
Is there a finite dimensional subcomplex of the Morse-Bott complex that computes the cohomology of a manifold?
In the paper Morse-Bott theory and equivariant cohomology, Austin and Braam built the Morse-Bott (geometric) complex which computes the de Rham cohomology of the manifold $M$ (this paper can be found ...
- 219
22
votes
3
answers
996
views
Equilaterally triangulated surfaces with prescribed boundary
There is a problem in Richard Kenyon's list (Wayback Machine) which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest ...
- 7,047
5
votes
2
answers
347
views
Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$
According to Theorem 1.7 of Mark-Tosun's paper, the Brieskorn sphere $\Sigma(2,3,6m+1)$ admits two tight contact structure $\xi_{i}\ (i=0,1)$. They are both Stein fillable and they are contactomorphic ...
- 1,049
2
votes
1
answer
121
views
On the realization of a quotient group
Let $P$ be a finite polyhedron and $N$ be a normal subgroup of $G=\pi_1 (P)$. It is known that there exists a covering space $(\tilde{P},p)$ so that $p_* \pi_1 (\tilde{P})=N$. It follows that for the ...
- 139
13
votes
1
answer
258
views
Example of ''annihilation'' of Seiberg-Witten Equation solutions
The proof that the Seiberg-Witten invariants of a 4-manifold $X$ with fixed Spin$^c$ structure really are invariant wrt the metric used to define them goes roughly as follows (for simplicity let $b_2^+...
- 313
5
votes
1
answer
330
views
Examples of Morse functions on unit tangent bundle of the sphere $T^1(S^2)$
In a sense, calculus is all about the study of critical points of functions on flat space $\mathbb{R}^N$ (e.g. here). Let's try a different venue, the unit tangent bundle of the sphere.
$$ T^1(S^2) =...
- 22.6k
7
votes
0
answers
396
views
Finding basis of cohomology of a symplectic manifold by using Symplectic Minimal Model Program
My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology.
Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...
user21574
19
votes
2
answers
1k
views
Is there a geometric interpretation for Reidemeister torsion?
Given a finite CW or simplicial decomposition of a space $X$ and a ring homomorphism $\varphi:\mathbb{Z}[\pi_1(X)]\to F$ for a field $F$, if the $\varphi$-twisted homology is trivial, then the ...
- 413