All Questions
Tagged with gt.geometric-topology reference-request
361 questions
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
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
2
votes
1
answer
400
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
2
votes
3
answers
511
views
Blaschke Condition for hyperbolic lattices
For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
- 975
2
votes
2
answers
305
views
boundary homomorphism in the homotopy exact seqeunce of principal $SO(9)$ bundle over $S^8$
Consider principal $SO(9)$ bundles over $S^8$.They are in 1-1 correspondence with $$[S^8,BSO(9)]\cong \pi_7(SO(9))\cong \mathbb{Z}$$
Now pick up one such bundle $\xi$,we have the long exact sequence ...
- 111
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
1
answer
102
views
Complexity of recognizing equivalent translation surfaces
"A translation surface is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence."
I take that succinct (and not fully precise) definition from a ...
- 151k
2
votes
2
answers
487
views
Some general properties of arithmetic groups of simplest type
I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...
- 2,436
2
votes
1
answer
532
views
Meaning of " Open Book cannot be Stabilized Further"?
I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.) ...
- 23
2
votes
1
answer
308
views
Hopf reference sought
For a vector $w$, let $T_{w}$ be the translation by $w$.
I was told that the following observation about subsets of the plane was due to H. Hopf:
Let $X$ be a compact, path-connected subset of the ...
- 101
2
votes
1
answer
400
views
References for the categories: DIFF PL LIP TOP
Is there any survey paper focusing on the study of DIFF PL LIP TOP categories?
- 1,101
2
votes
1
answer
188
views
Surveys on unknotting number
Any knot diagram could be converted to an unknot by cross change.
The unknotting number of a knot diagram is the minimal number of cross changes needed.
A knot could have many different diagrams and ...
- 91
2
votes
1
answer
111
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 ...
- 1,389
2
votes
2
answers
251
views
What are these compact sets called?
I'm wondering if a compact set $A\subset\mathbb{C}$ satisfying the properties that
• $A$ and its complement have finitely many connected components
• every connected component of $\partial A$ is the ...
- 5,428
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
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
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^...
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
2
votes
0
answers
218
views
Skein relation, Braids, and Hecke algebra
Many knot invariants (e.g. Alexander polynomial, Jones Polynomial,etc) admit a recursive algorithm
based on the so-called skein relation
But why the skein relation is a natural thing?
People have been ...
- 21
2
votes
0
answers
307
views
Theorem classifying fixed point sets of an isometry of the three sphere
Let $T:S^3\rightarrow S^3$ be an isometry of finite order. Then the set $S^T=\{x\in S^3|Tx=x\}$ of fixed points is either empty, or a pair of antipodal points, or a great circle, or a great sphere.
...
- 422
2
votes
0
answers
208
views
Retracting to a bigger compact
Consider the topological spaces $X$ with the following property:
For every compact $K\subseteq X$ there is a compact set $L$ such that $K\subseteq L\subseteq X$ and $L$ is a retract of $X$.
Let ...
- 128k
2
votes
0
answers
543
views
Tubular neighbourhoods are unique up to ambient isotopy?
Let $M$ be a closed smooth submanifold of $N$. It is well known that tubular neigbourhoods of $M$ are diffeomorphic to the normal bundle of $M$ in $N$ and therefore to each other. Are they smoothly ...
- 313
2
votes
0
answers
315
views
smooth structure on complete intersection
A complete intersection is an algebraic variety cut out by homogenous polynomials. Geometrically, this is the intersection of hypersurfaces in complex projective space.
Below, let's confine to the ...
- 59
2
votes
0
answers
214
views
Connecting homomorphism in generalized cohomology theory
I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence
$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial ...
- 21
2
votes
0
answers
200
views
Gluing two diffeomorphisms and then smoothing
This question did not get an adequate answer on math.stackexchange.
Let $M_1,M_2$ be two $n$-dimensional closed manifolds and suppose that $M_i=\bar{U}_i^+\cup \bar{U}_i^-$ where $\bar{U}_i^\pm$ are ...
- 1,149
2
votes
0
answers
110
views
Pure braid groups of the complement of a lattice in the complex plane: generators and relations
Where can I find a presentation (by `natural' generators and relations between them)
of the pure braid groups $PB_n(S)$ (for $n>0$) of $S=\mathbb C\setminus (\mathbb Z\oplus i \mathbb Z)$?
Thanks ...
- 838
1
vote
3
answers
502
views
orbit space of $\mathbb{Z}_p$ action over complex projective space by permuting the homogeneous coordinates
$Z_p$:=cyclic group of order $p$.
I want to understnd $H_\ast(\mathbb{C} P^{n}/Z_{n+1};Z)$ with $(n+1)$ being a prime number,and the action is given by permuting the homogeneous coordinates.
For ...
- 593
1
vote
2
answers
498
views
need a reference (topological manifolds)
hello, I need a book where i can find the proof for the classification of 1-dimensional topological manifolds.
(i already have Milnor's for the classification of 1-dimensional smooth manifolds)
thank ...
- 133
1
vote
1
answer
265
views
Boundary components of a subsurface
Consider the following situation. Suppose we have a closed oriented Riemannian surface $ \Sigma $ and a connected open subset $ \Omega \subseteq \Sigma $ with a boundary, consisting of finitely many ...
- 337
1
vote
2
answers
350
views
Commutativity in the Fundamental Group and Knot Theory
Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
- 1,108
1
vote
2
answers
336
views
The moduli space of finite volume hyperbolic 3-manifolds?
By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$.
I will call
$$\mathcal{M}...
- 223
1
vote
1
answer
124
views
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
My question is in the tittle:
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
If the answer is yes, is there a reference for this.
- 11
1
vote
1
answer
271
views
Ratner theorem and dense geodesic planes in hyperbolic manifolds
Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
- 829
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
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
1
vote
2
answers
262
views
Reference request for widely used theorem
I am looking for a reference to the theorem that any oriented closed surface of genus $g$ is a 2-fold cover of $S^2$ (branched over 2$g$+2 points).
- 31
1
vote
1
answer
773
views
Dehn twist generators for mapping class group of a genus zero surface with boundary
Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the ...
- 143
1
vote
1
answer
326
views
Addition of two homology classes is zero in construction of Poincare Sphere
I ask here the question since it hasn't been answered in
Math Stack Exchange.
I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
- 909
1
vote
1
answer
71
views
Terminology: Co-completion of Met?
In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, ...
- 5,405
1
vote
1
answer
287
views
Jones polynomial of cable knots
Let $K_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$.
Is there a closed formula for the Jones polynomial for $K_{p,q}$ as in the case of Alexander polynomial or Seifert matrices?
user160180
1
vote
1
answer
414
views
Equivariant maps inducing isomorphism in integral cohomology
Consider the following statement.
Suppose $X$, $Y$ are finite CW-complexes with free involution
and $\mu:X\to Y$ is an equivariant map.
If $\mu^*:H^i(Y;\mathbb{Z})\to H^i(X;\mathbb{Z})$ is an ...
- 11
1
vote
1
answer
379
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
1
vote
0
answers
170
views
Uniqueness of collar neighborhoods for non-compact boundary case in smooth setting
Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times
\partial M \to M$ be two smooth embeddings that are the identity map
on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is
...
- 265
1
vote
0
answers
154
views
Reference for hyperplane arrangements
I am interested in the hyperplane arrangement in $\mathbb{C}^n$ defined by the polynomial
$$
(x_i-x_j)(x_i+x_j)
$$
for $1<i<j\leq n$.
I vaguely recall that the completion of this arrangement ...
- 1,759
1
vote
0
answers
227
views
What does it mean for two natural numbers to be *approximately equal*?
This is related to this other question of mine about a paper of Colin and Honda.
I'm trying to follow the proofs line by line. I found the following piece of notation that is not explained in the ...
- 1,409
1
vote
0
answers
126
views
Flows commuting with Anosov flows and further reference request
Hello respected members of Mathoverflow. I was reading the paper "Flots d’Anosov dont les feuilletages stables sont différentiables" by Etienne Ghys and there was a statement which he remarked was ...
- 229
1
vote
1
answer
169
views
A proof of Edelstein and Kelly theorem
Edelstein and Kelly theorem states the following.
Let $A$, $B$ and $C$ be $3$ nonempty finite subsets of points in $\mathbb{R}^n$ such that affine-span $(A \cup B \cup C)$ has dimension at least $4$ ...
- 2,576
1
vote
0
answers
59
views
Integer valued signature of $4n$ dimensional orbifolds
Let $M^{4n}$ be a smooth oriented $4n$-dimensional manifold without boundary. Then we have an intersection form in $H^{2n}(M^{4n},\mathbb R)$ and such a form has signature $(n_+, n_-)$.
Question. I ...
- 14.3k
1
vote
0
answers
39
views
Homology of the subcomplexes of the "diamond shaped" sphere under 1-norm in $R^n$ as a simplicial complex
The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$.
The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...
- 459
1
vote
0
answers
561
views
The fundamental group of the complement of codimension 2 submanifold
Suppose that $M^n$, $V^{n+2}$ are connected, compact smooth manifolds.
Let $f\colon M^n\to V^{n+2}$ is a smooth embedding.
Let $K_f$ be the kernel of the inclusion induced homomorphism $\pi_1(V-f(...
- 11