# 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).

4,153
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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 ...

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84
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### 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 ...

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3
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### 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 $...

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48
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### 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 ...

5
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313
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### 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 $\...

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57
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### 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 ...

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2
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234
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### 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 $S$....

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216
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### 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 ...

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1
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108
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### 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 ...

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1
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76
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### 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 ...

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+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{...

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181
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### 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" ...

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160
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### 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 $Q$ ...

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### 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....

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### 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-...

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84
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### 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 ...

6
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1
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147
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### 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 ...

6
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1
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276
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### 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 ...

3
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218
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### 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 ...

13
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1
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533
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### 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 ...

4
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1
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169
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### 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} $...

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### 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 ...

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94
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### 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 $\gamma > 0$ ...

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1
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77
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### 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}_{...

8
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4
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556
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### 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-...

4
votes

1
answer

387
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### 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 ...

1
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0
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93
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### 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 $\partial N=M$, ...

3
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1
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118
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### 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 ...

2
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0
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386
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### $$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [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(...

3
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1
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165
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### 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?

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1
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48
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### 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 ...

5
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1
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136
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### 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 ...

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2
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219
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### 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 ...

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1
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160
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### 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 ...

6
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1
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116
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### 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 ...

2
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0
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130
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### 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 ...

6
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1
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291
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### 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 $Y\times\{1\}$, $K$...

3
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1
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219
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### 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 ...

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### 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\...

2
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1
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172
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### 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:
...

4
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146
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### 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 ...

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0
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61
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### 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 ...

4
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1
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111
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### 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 ...

4
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1
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179
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### 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,...

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0
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### 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 ...

2
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1
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### 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 ...

2
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1
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157
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### 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.
...

7
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### 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 ...

0
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1
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188
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### 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 $\...

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0
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82
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### 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 ...