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).

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The equation of cubic surface

I was studying the nature of singularities of cubic hypersurface on $\Bbb P^{n+1}$. The chosen form was $$f(x_0,x_1 \dots , x_{n+1})=x_0^3+x_1^3+\dotsb+x_{n+1}^3-c(x_0+x_1+\dotsb+x_{n+1} )^3=0.$$ I ...
mecid. s.'s user avatar
1 vote
0 answers
84 views

Calculation of the distance of cocycles (the telescope formula for the difference of the nth iterates)

Let $T: X \to X$ be a Lipschitz continuous on a compact metric space $(X, d_1)$. Assume that $Y$ is a Banach algebra and we consider the metric $d_2$ in the space $Y$. Let $f:X \to Y$ be a uniform ...
Adam's user avatar
  • 1,021
3 votes
3 answers
248 views

Fundamental group of a generalized connected sum

Let $M$ and $N$ be two $n$ dimensional connected closed manifolds with $n \ge 3$, and let $S$ be a $(n-1)$ dimensional closed submanifold common to $M$ and $N$. Consider the connected sum of $M$ and $...
TopologyStudent's user avatar
1 vote
0 answers
48 views

Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"

Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
rab's user avatar
  • 159
5 votes
1 answer
313 views

Linking number and intersection number

Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
user429294's user avatar
0 votes
0 answers
57 views

Geometric intuition behind hyper-sphere volume recurrence relation [closed]

There is a recurrence relation for calculating the volume of a hyper-sphere and a logical explanation for why it holds, as well as other explanations here on MO. Is there a geometric intuition behind ...
Hank's user avatar
  • 1
7 votes
2 answers
234 views

Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface

The question is simple: For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$....
one potato two potato's user avatar
0 votes
1 answer
216 views

Relationship between quotient CW-complexes after attaching cells

I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
William Thomas's user avatar
0 votes
1 answer
108 views

Continuous extensions of tangent vector fields

Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
MathLearner's user avatar
0 votes
1 answer
76 views

Continuous modification of tangent vector fields

Let $\Omega$ be an open subset of $S^2$, and assume that there exists a continuous tangent vector field $F(x)$ defined on $\bar{\Omega}\neq S^2$ with $|F(x)|=1$ for all $x\in \bar{\Omega}$. Suppose a ...
MathLearner's user avatar
2 votes
0 answers
158 views
+100

Homeomorphically deforming one continuous function to another

Motivation: A result of Klee Theorem 3.4 in [1], namely Klee's trick, implies that if $f,g:\mathbb{R}^d\to \mathbb{R}^n$ for any $d,n\in \mathbb{N}_+$ then there is a homeomorphism $\tilde{H}:\mathbb{...
ABIM's user avatar
  • 4,969
4 votes
1 answer
181 views

Residual finiteness and a gluing problem

The below flowchart is from Thurston's paper Hyperbolic structures on 3-manifolds I. I don't know if I interpreted it correctly but at the bottom it says that Residual finiteness "implies" ...
one potato two potato's user avatar
4 votes
0 answers
160 views

Obstruction to finding a Whitney disk

Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ ...
João Lobo Fernandes's user avatar
-2 votes
0 answers
95 views

Generalized writhe

I think that the Stasiak 3D writhe can be seen as a semi-invariant (Reidemeister 2 and 3, but not 1, i.e. regular isotopy) taking integer values (if measured in 4/7 units), just as the ordinary writhe....
Hauke Reddmann's user avatar
2 votes
0 answers
90 views

Are oriented-$h$-cobordant lens spaces orientation-preservingly homeomorphic?

Consider two three-dimensional lens spaces $N_1=L(p,q_1)$ and $N_2=L(p,q_2)$, and assume that there is an oriented-$h$-cobordism between them. In other words, we assume that there is an oriented four-...
Nathan's user avatar
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2 votes
0 answers
84 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 ...
Evan Scott's user avatar
6 votes
1 answer
147 views

Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle over a surface

In classic, Euler numbers associated to circle bundles over a fixed surface classify all possible such bundles. But the construction of Euler class in general requires the fact that any fiber bundle ...
one potato two potato's user avatar
6 votes
1 answer
276 views

Loop manipulation subgroup of the braid group

Recently, I came across a subgroup of the braid group $B_{2n}$ that I'm calling the "loop manipulation" group $H_n$. The idea is that we treat pairs of adjacent strands in the braid group as ...
pgadey's user avatar
  • 647
3 votes
0 answers
218 views

What is the image of a smooth map? [migrated]

Let $f: S^2 \to \mathbb{R}^n$ be a smooth map from the two-dimensional sphere to euclidean space. Let $X = \mathrm{Im}(f) \subset \mathbb{R}^n$ be the image topological space (note: the quotient ...
unknownymous's user avatar
13 votes
1 answer
533 views

Identifying two definitions of orientation on a vector space

Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$: A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
Jean's user avatar
  • 133
4 votes
1 answer
169 views

Dehn surgery on $RP^2 \times S^1$

A standard example of Dehn surgery is obtaining $S^3$ from $S^2 \times S^1$. Consider a unknot $L$ wrapping the non-trivial cycle $S^1$ in $S^2 \times S^1$. We drill out a tubular neighborhood $T_{L} $...
Topology_Dummy_Boy's user avatar
-4 votes
0 answers
77 views

Is 3d writhe for tight rational tangles quantized?

The 3d writhe of (simple) ideal (i.e. tight) knots is quantized, as Stasiak and colleagues have shown years ago. What's the current state of progress on the question whether the writhe of tight ...
Tangle's user avatar
  • 1
1 vote
0 answers
94 views

Mixing of geodesic flow and rate of mixing

I know the following result which is if $X:=G/\Gamma$ where $G$ be a Lie group and $\Gamma$ is a lattice in $G$. Then geodesic floe $F=(g_t)$ is exponentially mixing i.e., there exist $\gamma > 0$ ...
User1723's user avatar
  • 307
1 vote
1 answer
77 views

Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
  • 127
8 votes
4 answers
556 views

Residual finiteness of hyperbolic 3-manifold groups

So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is: Q1. If $M$ is an infinite-...
one potato two potato's user avatar
4 votes
1 answer
387 views

Hyperbolic three-manifolds that fiber over the circle

Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
user524868's user avatar
1 vote
0 answers
93 views

Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
Zhiqiang's user avatar
  • 881
3 votes
1 answer
118 views

The boundary regularity of a Teichmüller domain

By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is ...
Mahdi Teymuri Garakani's user avatar
2 votes
0 answers
386 views

$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
  • 625
3 votes
1 answer
165 views

Principal circle bundles over punctured $3$-sphere

Let $M$ be $S^3$ with $k$ disjoint open balls $D^3$ removed. Can we classify all principal circle bundles over $M$ such that the total space is simply-connected?
Zhiqiang's user avatar
  • 881
0 votes
1 answer
48 views

Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K? [closed]

Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components. Let $U$ be a connected component of $S \setminus K$ and ...
Fernando Oliveira's user avatar
5 votes
1 answer
136 views

Heegaard splitting of figure-8 knot complement

It is well-known that the figure-8 knot complement in $S^3$ can be described as a circle fibration of a once-punctured torus. Is there also a description of the figure-8 knot as a Heegaard splitting ...
Oblonski's user avatar
  • 103
3 votes
2 answers
219 views

For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
Learning math's user avatar
2 votes
1 answer
160 views

Order of a loop around a cone point

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
RKS's user avatar
  • 533
6 votes
1 answer
116 views

Knotted concordances of slice links

Are there any examples of a link $L$ such that: $L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap ...
Alessio Di Prisa's user avatar
2 votes
0 answers
130 views

Mutants or not?

Two 4-tangles (drawn unneccesarily complicated to show how they are related - both are 6-tangles capped off with the same cap): (alternate version with ends at the same point) If I could turn over ...
Hauke Reddmann's user avatar
6 votes
1 answer
291 views

Slice knots in 3-manifolds

Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$...
Qiuyu Ren's user avatar
3 votes
1 answer
219 views

Rational 4-tangles vs rational knots

The closure of a rational $4$-tangle is a rational knot. But is the converse true? We could tangle up even the unknot to a hopeless mess before cutting it up, and we could cut it were it "hurts ...
Hauke Reddmann's user avatar
9 votes
0 answers
216 views

Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces

Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense? In other words, when does at set $S\...
Christian's user avatar
  • 779
2 votes
1 answer
172 views

Example of pseudo $3$-manifold without any shape structure

I'm reading Andersen and Kashaev's A TQFT from quantum Teichmüller theory and the following condition in their definition of admissible oriented triangulated pseudo $3$-manifold confused me: ...
Shana's user avatar
  • 237
4 votes
0 answers
146 views

Subdivision via poset maps and pullback

In the following, all posets and complexes are assumed to be finite. For a poset $P$ denote by $|P|$ its geometric realization or nerve (i.e. forming the order complex and taking its geometric ...
KoopaTroopa's user avatar
1 vote
0 answers
61 views

Clique complex of expander graphs simply connected?

Given an expander graph family (an injective sequence of graphs with uniformly bounded vertex degree and a Cheeger constant/Laplacian spectral gap uniformly bounded away from zero). Can the ...
Florentin Münch's user avatar
4 votes
1 answer
111 views

Rigidity/flexibility of Sol-structures on closed 3-manifolds

This is a follow-up to the question Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds From the answers/comments there and from an excellent survey by Bonahon ...
Roman's user avatar
  • 173
4 votes
1 answer
179 views

For which quadratic number field, the algebraic integers are cusps for some Coxeter group?

Let $H^2=\{(x,y)\mid y>0\}$ be the hyperbolic upper-half plane. Let $K=Q(\sqrt{d})$ be a quadratic number field, and $\mathcal{O}_K$ be the ring of algebraic integers in it. Let $\Gamma=\Delta(p,q,...
zemora's user avatar
  • 545
1 vote
0 answers
69 views

Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
53Demonslayer's user avatar
2 votes
1 answer
91 views

Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...
stupid_question_bot's user avatar
2 votes
1 answer
157 views

Guts of 3-manifolds for sutured manifolds and pared manifolds

I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me. ...
Fredy's user avatar
  • 492
7 votes
1 answer
176 views

Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds

It is known that closed spherical and hyperbolic 3-manifolds are rigid. I.e., if two such manifolds are diffeomorphic, then they are isometric (moreover, I think, that every diffeomorphism is isotopic ...
Roman's user avatar
  • 173
0 votes
1 answer
188 views

Fixed points free automorphisms of Teichmüller spaces

Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
Mahdi Teymuri Garakani's user avatar
1 vote
0 answers
82 views

Fibre product and submersion of PL-manifolds

Let us consider the fibre product $M\times_{f,g} M'$ of $M \xrightarrow{f} N \xleftarrow{g} M'$. If $M,M'$ and $N$ are smooth manifolds and $f$ is a submersion, then $M\times_{f,g} M'$ is again a ...
KoopaTroopa's user avatar

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