# 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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107 views

### Thurston universe gates in knots: which invariant is it?

Today I discovered this nice video of a lecture by Thurston: https://youtu.be/daplYX6Oshc in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
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### Surface separating the boundary of a cylinder

Let $M^2$ be a connected closed surface. Suppose there exists an smooth embedding from a connected closed surface $N$ into the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and ...
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### Smooth Schoenflies Theorem for compact $3$-manifolds

Let $M^3$ be a compact $3$-manifold with $\partial M=N$ a connected surface. Suppose one has a smooth embedding of $N$ into the interior of $M$ and $N$ bounds a domain $D$ in $M$. Can we show that $D$ ...
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### Surface in a product domain

Let $M$ be a connected closed surface. Suppose $N$ is a connected closed surface embedded in the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and $M \times \{1\}$. Can we ...
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### 3-dimensional h-cobordisms

Let $W$ be a $3$-dimensional $h$-cobordism of closed surfaces $M_0$ and $M_1$. Can we prove that $W$ is trivial? That is, $W$ is homeomorphic to $M_0 \times [0,1]$.
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### Mapping class groups of Haken Seifert 3-manifolds (not small)

This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $M$, in particular how to systematically compute it. I want to ...
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### Suspension of a topological space

Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
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### Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?

The question is, for a smooth embedding $$f : S^3 \to S^2 \times D^3$$ one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$. Which ...
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### Realizing groups as the fundamental group of graphs of groups allowing non-injective maps?

In the definition of a graph of groups it is assumed that the maps from the edge groups to the vertex groups of injections, however in what follows I will also be interested in the case where the maps ...
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### Books in advanced differential topology

I am looking for books or other sources in differential topology that include topics like: vector bundles, fibration, cobordism, and other related topics. In general, if anyone has recommendation of ...
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### Can every manifold be represented as a quotient

My question is "inspired" by the uniformization theorem for Riemmannian surfaces and this post. Suppose that $X$ is connected (finite-dimensional) topological manifold without boundary. ...
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### Composite knots and their braid words

Given a composite knot K = K_1 # K_2, I wonder how the braid word looks like. Is it possible to see from the word that the knot is composite? I am not aware of a statement such as "the closure of ...
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### Do different decompositions of Dehn twists induce the same cobordisms between mapping tori?

Suppose $Y=T^2\times I/f$ is a mapping torus, where $f$ is a diffeomorphism of $T^2$. Suppose $c$ is a curve on $T^2\times \{1\}$ with surface framing and suppose $D_c$ is the positive Dehn twist ...
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### Hyperbolization with word-hyperbolic fundamental group

In Davis-Januszkiewica´s paper Hyperbolization of polyhedra it is shown that for every manifold $M$ there exists a map $N \to M$ of non-zero degree such that $N$ is aspherical (plus some more ...
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### Compactly supported geometric representatives for Seiberg-Witten invariant

The question is introduced at the end of the second paragraph. Readers familiar with Seiberg-Witten theory may well skip the first paragraph. The first paragraph is meant to set up some notation which ...
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### Characterizations of metric trees

Let $X$ be a geodesic space. Then the following conditions are equivalent: For any $x,y\in X, x \neq y$, there is a unique arc (homeomorphic to the interval $[0,1]$) with endpoints $x$ and $y$. No ...
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### Presentations of mapping class groups in dimension $3$

For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then ...
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### Approximate Jordan-Brouwer theorem

This came up when thinking about this question. It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is ...
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### What do you call this notion of similarity between subsets of a metric space?

Given a metric space $X$, we say a continuous isometry is a continuous map $F:X\times[0,1]\to X$ such that defining $f_t(x) = F(x,t)$ for $t \in [0,1]$, we have that $f_0$ is the identity map and $f_t$...
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### Functoriality of Thurston's norm

Let $M$ be a manifold of dimension $3$ and let $N$ be an embedded submanifold of $M$ (also of dimension $3$). Then, both second homologies $H_2(M)$ and $H_2(N,\delta N)$ are equipped with a norm (...
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### PL-embeddings of balls into PL-manifolds

Let $B$ be $k$-dimensional PL-ball and let $M$ be a connected $n$-dimensional PL-manifold, let's say without boundary. Furthermore let $f,g\colon B\to M$ be two PL-embeddings. If $k=n$, then the disc ...
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### Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?

There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (...
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### Restrictions on pointed lifts of isometries

Let $M$ be a (closed) Riemannian manifold and let $f$ be an isometry of $M$ fixing a point $\ast \in M$ that acts trivially on $\Gamma := \pi_1(M,\ast)$. Then there is a unique isometry $\tilde{f}$ of ...
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### Exotic analytic triangulations of $S^5$?

I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf http://www-math.mit.edu/~...
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### Ideal Ford domain (for finite index subgroup)

Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices $g= \begin{pmatrix} \alpha & \overline{\beta} \\ \beta & \overline{\alpha} \end{pmatrix}$...
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### Ping pong with parabolic isometries on Gromov hyperbolic spaces

For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
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### The Kirby diagram of a manifold glued along the lens space $L(p,1)$

Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
### Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence
Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to any self-homotopy ...