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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1answer
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 ...
3
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1answer
86 views

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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1answer
44 views

Compatibility of two cylindrical regions

Let $M^2,N^2$ be connected closed surfaces. Suppose there exists region $D$ in the interior of $M \times [-2,2]$ such that (a) $D$ is homeomorphic to $N \times [0,1]$; (b) $D$ contains $M \times [-1,1]...
2
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2answers
84 views

Embedded submanifold in a cylinder

Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $...
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0answers
87 views

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 $D$ ...
1
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1answer
62 views

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 ...
6
votes
1answer
166 views

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 $M_0 \times [0,1]$.
4
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1answer
89 views

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 ...
5
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0answers
91 views

Computation of partial configuration spaces ala Cohen

Let $F_i$ denote the subset of $\mathbb{R}^n$ consisting of tuples $(x_1,\dots,x_n)$ where there are less than or equal to $i$ unique entries. Is there a computation of the homology of $\mathbb{R}^n - ...
4
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0answers
202 views

What field of mathematics is this? Necessary and sufficient corridors for topological routing

I am a computer scientist working on a problem in electronics design. The overall problem is about how to route traces on a circuit board, and I am looking for help on one of the subproblems. We ...
4
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1answer
92 views

Concordance classes of diffeomorphisms of $D^4$

Is anything known about the set of concordance classes (also called pseudoisotopy classes) of the relative to the boundary diffeomorphisms of $D^4$?
3
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1answer
71 views

Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper (https://arxiv.org/pdf/math/0510312). Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
3
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0answers
113 views

Commutator length of the fundamental group of some grope

A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra $L_0 \to L_1 \to L_2 \to \cdots$ obtained as follows. Take $L_0$ as some $S_g$, an ...
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0answers
81 views

Open cone homeomorphic to the Euclidean space

Let $X$ be a topological space and the open cone $C(X)$ over $X$ is defined to be $X \times [0,1)$ with $X \times \{0\}$ identified. Suppose $C(X)$ is homeomorphic to $\mathbb R^4$, can we prove that $...
17
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2answers
1k views

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?
23
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2answers
607 views

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 ...
4
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1answer
115 views

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 ...
11
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6answers
1k views

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 ...
4
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1answer
322 views

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. ...
5
votes
1answer
92 views

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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0answers
59 views

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 ...
2
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1answer
85 views

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 ...
4
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0answers
138 views

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 ...
8
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1answer
229 views

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 ...
9
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2answers
244 views

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 ...
3
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0answers
90 views

When is the dual block decompositition a CW decomposition?

Given a triangulated homology manifold $X$, the dual block decomposition is defined by setting, for each simplex $\sigma$ of $X$, the block $\overline{D}(\sigma)$ to be the union of all simplices of $\...
3
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0answers
152 views

Mapping class group of a twisted I-bundle over $RP^2$

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Homeo{Homeo}$Let $\Mod(M)=\pi_0(\Homeo(M))$ be the mapping class group of a manifold, possibly with boundary (I'm including the orientation reversing ...
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1answer
93 views

Approximate Jordan-Brouwer theorem (corrected)

My first attempt to ask this question sort of failed (I'll explain below). This came up when thinking about this question. It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{...
2
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1answer
111 views

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 ...
1
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0answers
66 views

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 $f_t$...
4
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1answer
195 views

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 (...
8
votes
2answers
197 views

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 ...
3
votes
0answers
87 views

Logarithm on formal power series continuous?

Denote $V:=\mathbb{R}^d$ and consider the Cartesian product $V^\infty:=\prod_{k=0}^\infty V^{\otimes k}$ together with its canonical projections $\pi_k : V^\infty\rightarrow V^{\otimes k}, (v_0, v_1, \...
3
votes
1answer
147 views

Does an amphichiral knot admit roughly twice as many knot diagrams of a given crossing than a chiral knot?

Recall that a knot is amphichiral (or achiral) if there is a continuous deformation of the knot into its mirror image. I'm interested in knowing when and whether we can use approaches like Stockmeyer ...
4
votes
0answers
102 views

Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?

I am working with manifolds with boundary pattern, as defined by Matveev in his book Algorithmic Topology and Classification of 3-Manifolds. A manifold with boundary pattern is a pair $(M, P)$ where $...
1
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1answer
116 views

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 (...
3
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0answers
97 views

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 ...
10
votes
0answers
429 views

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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0answers
40 views

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} $...
3
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0answers
104 views

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 ...
10
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0answers
403 views

Topological dimension, Hausdorff dimension, and Lipschitz mappings

I can prove the following result. Here $\operatorname{dim} X$ stands for the topological dimension and $\mathcal{H}^n$ denotes the Hausdorff measure. Theorem. Suppose that $f:\mathbb{R}^n\supset\...
5
votes
1answer
308 views

$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?

Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion? My understanding so far — An $\...
2
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0answers
81 views

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 ...
1
vote
1answer
181 views

Partitioning a convex $n$-polygon

Let $P$ be a convex polygon with $n\ge3$ vertices. Let $\mathcal{Z}_K(P)$ be a partition of the polygon into $K$ polygonal (but not necessarily convex) parts whose interiors are pairwise disjoint, $$ \...
7
votes
2answers
227 views

Index of the mapping class group $\Gamma_{g,n}$ inside $\text{Out}(\Pi_{g,n})$

Let $\Sigma_{g,n}$ denote an $n$-punctured surface of genus $g$, with $2g+n-2 > 0$. Let $\Pi_{g,n}$ be its fundamental group (for some choice of base point), and let $\Gamma_{g,n}$ denote its pure, ...
1
vote
1answer
123 views

Action of the permutation group on the set of topologies on a continuum

Let $X$ be a set of continuum cardinality. The group of permutations of $X$ acts on the set of topologies on $X$. What can be said about the fixed points of this action? Are manifolds fixed points? ...
18
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2answers
482 views

Which knots appear as the singular locus of a polyhedral metric on the 3-sphere?

What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm ...
1
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0answers
68 views

Given a homeomorphism on $\mathbb{R}^3$, can its effects on a compact subset be realized by a homeomorphism that's non-identity only on a compact set?

Let $f_1 \colon \mathbb{R}^3 \to \mathbb{R}^3$ be a homeomorphism, and let $K_1 \subseteq \mathbb{R}^3$ be compact. Does there always exist a homeomorphism $f_2 \colon \mathbb{R}^3 \to \mathbb{R}^3$ ...
2
votes
0answers
96 views

Existence of a certain Morse function

Let $\mathbb{B}^4=\{(x_1, x_2, x_3, x_4): x_1^2+x_2^2+x_3^2+x_4^2\leq 1\}$ be the closed unit 4-ball. Consider first the Morse function $$ f(x_1,x_2,x_3,x_4)=\frac{1}{2}\left(x_1^2+x_2^2-x_3^2-x_4^2\...
8
votes
2answers
138 views

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 ...

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