All Questions
Tagged with gt.geometric-topology reference-request
361 questions
11
votes
2
answers
518
views
Knots having the same Alexander module which are not S-equivalent
As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules.
There are examples of knots ...
- 548
4
votes
0
answers
154
views
Is there a notion of "locally flat" for CW complexes?
A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\...
- 13.6k
4
votes
1
answer
169
views
"Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$
Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices).
I ...
- 13.6k
2
votes
0
answers
100
views
Equivariant disk theorem in dimension 2
All groups I'll consider are finite.
An important part of equivariant differential topology in dimension 3 is the equivariant disk theorem, which says that for a $G$-action on a compact $3$-manifold ...
- 21
7
votes
1
answer
292
views
Computing $\pi_1$ of the complement of a non-singular plane curve
The following is a well-known fact:
Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.
This was further ...
- 10.9k
12
votes
1
answer
747
views
Isotopic diffeomorphisms of the sphere
Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f$...
- 28k
2
votes
0
answers
194
views
Stable homeomorphism theorem and the annulus theorem
Brown and Gluck [BG] proved in 1964 that the stable homeomorphism conjecture implies the annulus conjecture.
Is the proof of this implication difficult?
Is there any other place with the proof of ...
- 28k
3
votes
0
answers
227
views
Classifying spaces beyond CW complexes
We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
- 803
2
votes
1
answer
365
views
Correspondence between fundamental group and geometric properties of $X$
At the time of studing some algebraic topology I was wondering about the following.
Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group.
If we assume some algebraic property of $\...
- 613
2
votes
0
answers
146
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
5
votes
1
answer
188
views
Properly embedded surfaces in handlebodies are compressible or boundary compressible?
I've read in a couple of different places (a paper and a blog) the following fact:
if $F$ is a surface, properly embedded in a three-dimensional handlebody of genus at least two, then $F$ is either ...
- 421
16
votes
3
answers
942
views
Maximal degree of a map between orientable surfaces
Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.
What is an elementary proof of ...
- 1,095
1
vote
1
answer
256
views
Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory
As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
- 131
2
votes
1
answer
144
views
English version of a paper by Gusarov
I am looking for the english translation of the paper in russian Variations of knotted graphs, geometric technique of n-equivalence, St. Petersburg Math. J. 12-4 (2001) by Gusarov.
There is a .ps file ...
- 23
11
votes
2
answers
317
views
Reference request - Fibrations between spaces of embeddings
This is a cross-post of this question from MSE.
Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
- 2,292
6
votes
1
answer
328
views
Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?
Question: Does $\overline{\mathbb{CP}^2 \setminus B^4}$ (that is the closure of complex projective plane minus a 4-ball) embed (smoothly/topologically/piecewise linearly) in $\mathbb R^5$?
Background: ...
- 976
0
votes
2
answers
309
views
Reference for a topological result
I am reading the short paper due to Erdös and Bollobás "On a Ramsey-Turán type problem", where they obtain a lower bound for the number of edges on an $n$-graph without $K_4$ as a subgraph ...
- 1,561
6
votes
1
answer
479
views
Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?
I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of ...
- 6,917
6
votes
1
answer
500
views
A characterization of metric spaces, isometric to subspaces of Euclidean spaces
I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$:
Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
- 41.8k
4
votes
0
answers
182
views
Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
- 41.8k
4
votes
1
answer
260
views
Bounds for the crossing number in terms of the braid index?
Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
- 9,114
3
votes
1
answer
235
views
Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,097
15
votes
1
answer
954
views
Extending diffeomorphisms
Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary.
Question. Is it possible to ...
- 28k
4
votes
2
answers
355
views
Books for learning branched coverings
I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create ...
- 613
8
votes
1
answer
281
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
- 1,097
3
votes
1
answer
212
views
Picturing twisting of strands explicitly after blow downs
In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
- 213
5
votes
0
answers
289
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, $V_2$, ...
- 17.6k
5
votes
1
answer
283
views
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
3
votes
0
answers
463
views
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,...
- 613
5
votes
1
answer
291
views
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
9
votes
1
answer
519
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 ...
- 1,095
5
votes
1
answer
151
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
6
votes
2
answers
395
views
Dual surfaces of a first cohomology class of a 3-manifold
Let $M$ be closed 3-manifold and $\alpha\in H^1(M;\mathbb Z_2)$ an arbitrary element. (In my case we know that $M$ is non-orientable and $\alpha^3=0$.) It is well known that there is a closed 2-...
- 1,095
1
vote
1
answer
378
views
Bridges between geometry and combinatorics
Geometry and combinatorics are two different branches of mathematics. Does there exist any connection between them? In many cases, mathematicians solve some geometric problems by reducing them to a ...
- 613
5
votes
0
answers
131
views
Earliest known proof of "Any degree one self-map of an orientable connected finite-type non-compact surface is homotopic to a homeomorphism"
I attended a talk where the speaker said the following is due to Nielsen. I searched here and there but couldn't find the corresponding paper, if any. So, what is the earliest known proof of the ...
- 1,097
3
votes
1
answer
375
views
Reference for triangle groups
Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
- 613
3
votes
0
answers
88
views
Is the thickening of a PL 2-disc in $\Bbb R^4$ a 4-ball?
Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski ...
- 13.6k
4
votes
0
answers
183
views
In how far does the Whitney trick work in the piecewise linear setting in $\Bbb R^4$?
I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will ...
- 13.6k
9
votes
1
answer
296
views
The works of González-Acuña and Duchon from 70s and 80s
I would like to access the following two works of González-Acuña from 1970:
González-Acuña, F. Dehn’s construction on knots. Bol. Soc. Mat. Mexicana (2) 15 (1970), 58–79.
and
González-Acuña, F. On ...
- 1,292
3
votes
1
answer
265
views
Classification of degree one map between two closed orientable surfaces without using induction on the genus
A theorem of Edmonds (see Theorem 3.1. of "Deformation of Maps to Branched Coverings in Dimension Two") says that
Theorem 1: A degree-one map between closed orientable surfaces is homotopic ...
- 1,097
7
votes
2
answers
366
views
Boundary of a $4$-manifold and the fundamental group
I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$,
Assume $\pi_1(N)$ is known,...
- 1,177
3
votes
3
answers
259
views
Reference for an easy lemma on homeomorphisms of connected manifolds
If M is a connected manifold of dimension $\geq 2$ then the set of orientation preserving homeomorphisms of M that are isotopic to the identity acts $n$-transitively on M for all positive $n\in\mathbb{...
- 1,517
2
votes
1
answer
205
views
Is every finitely generated classical Schottky group quasifuchsian?
$\DeclareMathOperator\PSL{PSL}$(Classical, finitely generated) Schottky groups are groups generated by finitely many hyperbolic elements of $A_i\in \PSL(2,\mathbb{C}), $ $i<n$ such that the ...
- 441
2
votes
0
answers
65
views
Connection between a function and its usage in geometry [closed]
I know nothing about geometry, but I found a function which seems to have something to do with geometry.
This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$
where $x,y,z$ is ...
- 121
0
votes
1
answer
83
views
Construct a homeomorphism of a surface that sends a subsurface to another subsurface
Let S be a compact orientable surface. Let A and B be two subsurfaces of S that have the same signature. How to check if there is a homeomorphism of S that sends A to B and if so, find one?
Here a ...
user127837
-1
votes
1
answer
263
views
Construct a homeomorphism between two surfaces
Given two compact oriented surfaces that have the same number of genus and boundary components. How to construct a homeomorphism that sends one to another?
user127837
3
votes
1
answer
190
views
Stabilizer of the action of the mapping class group on a pants decomposition
$\DeclareMathOperator\Mod{Mod}$Let $S$ be a surface and $P=\{a_1,...,a_n\}$ be a pants decomposition of $S$. Denote by $\Mod(S)$ the mapping class group of $S$.
Define the stabilizer of $\Mod(S)$ on $...
user127837
1
vote
1
answer
210
views
Liouville property of hyperbolic spaces
It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am ...
- 1,407
4
votes
1
answer
577
views
Reference request for Poincaré–Lefschetz duality as an intersection pairing
I believe the following is well known after talking to some experts, but I am unable to find a reference for the case with boundary.
Fix a field $F$ and an oriented $n$-manifold $(M,\partial M)$. We ...
- 5,839
2
votes
1
answer
399
views
Smoothness of Minkowski functional is equivalent to smoothness of boundary
Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$,
$$
f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\},
$$
is $C^1$ ...
- 5,405