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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3answers
2k views

What is the geometric shape of the Monster sporadic group?

Conway made the comment that the Monster group represents the symmetries of a shape in 196,883 dimensions, something like a "star you hang on a Christmas tree." My question is, What do we know (or ...
5
votes
1answer
245 views

Hyperbolic manifolds with infinite cyclic fundamental group

It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...
4
votes
2answers
459 views

How to compute fundamental groups of closed surfaces without using Van-Kampen theorem?

Denote $X=mP^2$ the sphere glued with $m$ Mobius bands. It has a polygon representation $a_1a_1...a_ma_m$, i.e. it's a quotient by a $2m$-sides polygon $P$. Let $o$ be the center point of $P$, $x_0$ a ...
2
votes
1answer
111 views

Does tangle closure determine the triviality of the tangle?

Let $NS(T), D_+(T), D_-(T)$ denote closures of a $2$-tangle $T$ as in the picture. $T_0$ (below) is called trivial. Question: Suppose that $NS(T)$ and $D_+(T)$ are trivial knots. Does it imply that $...
7
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0answers
153 views

Are triangulations of n-dimensional manifolds determined by lower-dimensional skeleta?

Suppose that $M$ is an $n$-dimensional manifold equipped with a triangulation $T$. Given $n\ge 1$, in order to recover $T$ (up to an isomorphism of simplicial complexes) one needs to know at least ...
19
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1answer
393 views

Can every manifold be dominated by a parallelizable one?

A closed, oriented $d$-manifold $M$ is said to dominate another such manifold $N$ if there exists a map $M \to N$ of non-zero degree. (This notion should not be confused with the unrelated concept of ...
11
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1answer
148 views

Does there exist a non-zero signed finite borel measure which is zero on all balls?

Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
5
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0answers
131 views

$\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence

I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the ...
3
votes
2answers
261 views

On what proper Gromov hyperbolic space does a free product act?

Per Bowditch, a group is relatively hyperbolic if it acts geometrically finitely on a proper geodesic Gromov hyperbolic space. A free product of two (or finitely many) finitely generated groups is ...
7
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1answer
231 views

Understanding fundamental group of Poincare homology sphere

I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. The dashed line denotes a trefoil whose tubular ...
4
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0answers
103 views

Controlling the intersection of two surfaces in $\mathbb{R}^3$

Let $F_1,F_2$ be two closed orientable surfaces embedded in $\mathbb{R}^3$ with genus $2g_1, 2g_2$, respectively (edit: with $g_1, g_2 \geq 1$). Is it possible to isotope around $F_1$ and $F_2$ so ...
7
votes
1answer
221 views

Thickness and hierarchical hyperbolicity

Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here. I've heard that it is open ...
4
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0answers
158 views

Smoothability of open 4-manifolds

F. Quinn proved that any open topological 4-manifold admits a smooth structure in Ends of maps III: dimensions 4 and 5. He first proves the generalized annulus conjecture: Suppose $h:D^j\times \...
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0answers
93 views

Boundary map for Mayer-Vietoris sequence for Bordism

I am trying to reproduce some of the details in how the Mayer-Vietoris sequence for bordism should go, especially in showing exactness using this definition of the boundary operator. I have tried to ...
10
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1answer
156 views

Free product decompositions of the fundamental group of Hawaiian Earrings

This is a spin-off of my question here, separated from the older question following Jeremy's suggestion. Definition. Call a group $G$ essentially freely indecomposable if in every free product ...
2
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0answers
86 views

Full automorphism group of a Bruhat-Tits building

If we start with a semisimple algebraic group $G$ defined over a non-archimedean local field and want to understand the relationship of this group with the full type-preserving automorphism group of ...
3
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0answers
69 views

Approximative extension of the autohomeomorphism of the complement of the trivial knot?

Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...
7
votes
1answer
193 views

Resolution graphs in the sense of Némethi

The following definitions are from lecture notes of Némethi. A surface singularity $(X,0)$ is defined by $$(X,0) = (\{ f_1 = \ldots = f_m=0 \}) \subset \mathbb (\mathbb{C}^n,0),$$ where $f_i : (\...
4
votes
0answers
159 views

Contractibility and orientation double cover

Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the ...
5
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0answers
106 views

Lattices of minimal covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$

What are the (uniform/non-uniform) irreducible lattices of minimal (or even small) covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$? Context: Such a lattice will ...
4
votes
2answers
204 views

Finite models for torsion-free lattices

Let $G$ be a real, connected, semisimple Lie group and $\Gamma < G$ a torsion-free lattice. Then does there exist a finite $CW$-model for $B\Gamma$? I know this to be true in many instances (e.g. ...
5
votes
1answer
180 views

3-balls with the same boundary in $S^4$ differ up to diffeomorphism

I am looking at this recent paper by Budney and Gabai and I am confused by a certain sentence in it. Theorem 4.7 states that if $\Delta_1$ and $\Delta_2$ are two 3-balls smoothly embedded in $S^4$ ...
4
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0answers
97 views

program to compute hurwitz numbers

Is there a computer program available to compute Hurwitz numbers easily? In fact I only care about counting covers $C\to\mathbb{P}^1$ branched over $0,1,\infty$, and am even willing to restrict to the ...
12
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2answers
799 views

Cobordism and Kirby calculus

It may be a simple question but I wonder to ask: Is it possible to draw a homology cobordism between $3$-manifolds by using the techniques of Kirby calculus? At least, for instance, Brieskorn ...
8
votes
2answers
400 views

On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$

In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds. $$\...
1
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2answers
267 views

Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$ [closed]

Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It ...
10
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1answer
227 views

Kirby calculus on Mazur manifolds

I have questions about Akbulut and Kirby's paper Mazur manifolds. I couldn't figure out the following equality passages: Any help will be appreciated.
11
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2answers
653 views

Acyclic group and finite CW-complex

Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ?
5
votes
1answer
126 views

Lengths of closed geodesics on a flat vs hyperbolic punctured torus

Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$. Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f:...
1
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0answers
52 views

Conjugation map on $\mathbb CP^n\smallsetminus \mathring{D}^{2n}$ that preserves boundary, $n\geq 4$ even

The conjugation map $$\mathrm{conj}\colon\mathbb{C}P^{n-1}\to \mathbb{C}P^{n-1}$$ extends to an endomorphism of $\mathbb CP^n\smallsetminus \mathring{D}^{2n}$ (unique up to homotopy), since $\mathbb ...
2
votes
1answer
118 views

The isometric immersion of a positively curved projective plane in 3-dimensional Euclidean space?

In 1903, W. Boy showed that the real projective plane $\mathbb{R}P^2$ can be immersed in the Euclidean space $\mathbb{E}^3$ (see Werner Boy, Math. Ann. 57 (1903), no. 2, 151-184.). Suppose a ...
3
votes
1answer
135 views

The plumbing graphs of Brieskorn spheres

Let $p,q$ and $r$ be positive integers. A Brieskorn sphere is a closed oriented $3$-manifold defined by $$\Sigma(p,q,r) = \{ x^p+y^q+z^r=0 \} \cap S^5.$$ Its fundamental group is well-known due to ...
5
votes
1answer
126 views

Applications of quantum representations of the mapping class group to quantum computers

Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in and 1 and 2. The following sources 3 ...
2
votes
2answers
314 views

Alexander duality and homology equivalence

While reading the paper of Kauffman and Taylor "Signature of links" I found the following situation. In the proof of Theorem 2.6 they suppose that two links $L_1, L_2\in \mathbb{S}^3$ are ...
8
votes
3answers
509 views

Heegaard splittings of Brieskorn spheres

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the ...
4
votes
0answers
91 views

An upper bound for second type of Reidemeister move

Suppose there are tow diagram $D_1$ and $D_2$ of knot $K$ with $c_1$ and $c_2$ crossing. Are there any bound of second type of Reidemeister move in term of $c_1$ and $c_2$? In other words, Are there ...
5
votes
1answer
202 views

Multisignature and homeomorphism type

In classical surgery theory, there is a map $$L_{n+1}(\pi_1M)\to S(M^n)$$ Element in $L_{n+1}(\pi_1M)$ is realized as surgery obstruction of a surgery problem to $M\times I$ with one boundary piece ...
0
votes
0answers
76 views

Smooth immersed surfaces with an unknot as the boundary in $S^3$

Starting from a smooth embedded disk with an unknot boundary in $S^3$, if we make the boundary unknot complicated by performing an arbitrary but finite number of Reidemeister moves & planar ...
12
votes
1answer
503 views

Classifying space for Thompson's group F?

Let $\mathcal C$ be the free monoidal category generated by an object $X$, and a morphism $X \otimes X \to X$. This category contains exactly two connected components: that of the monoidal unit $1\in ...
9
votes
2answers
421 views

Existence of fibered surfaces in arbitrary 4-manifolds?

It is apparently a result of F. González-Acuña that all closed orientable 3-manifolds contain a fibered knot. (I am not sure exactly where to find a published proof of this result and as an aside I ...
8
votes
0answers
165 views

Dualizing module for $\operatorname{Aut}(F_n)$

In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...
3
votes
1answer
187 views

Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space

Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂𝑋 be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich https://...
7
votes
0answers
262 views

Self diffeomorphism of $S^2\times S^2$

The main question is motivated from the answer of this question https://math.stackexchange.com/questions/2481200/finite-groups-gs-which-acts-freely-on-s2-times-s2 Is it true that every self ...
8
votes
1answer
318 views

Topological mapping class groups of 4-manifolds

It is a classical result of Quinn that for a simply-connected closed $4$-manifold $X$ the isometries of its intersection form are in one-to-one correspondence with $\pi_0 \text{Homeo}(X)$. (Isotopy ...
7
votes
0answers
124 views

Long non-deformable hyperbolic fillings

The title and question have been edited in light of Ian Agol's comment. The previous question was stated in terms of the wrong notion of length to discuss deformations: What is the longest slope $\...
3
votes
0answers
135 views

Hochschild cohomology of (generalizations) of Khovanov's arc algebra

Backgroud: In his seminal paper A functor-valued invariant of tangles, Khovanov (among many other things) introduced the arc algebra $H^{n}$ and several functors between $H^{n}$ and $H^{m}$ related to ...
14
votes
0answers
176 views

Hauptvermutung for non-manifolds

The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement. People are mostly interested ...
9
votes
0answers
231 views

Finite quotients of surface braid groups

Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$. There is a braid $A_{12} \in \Sigma_b$ ...
9
votes
2answers
665 views

A plausible hyperbolic link

This link is hyperbolic according to SnapPy's computation. There is an obvious non-boundary parallel annulus spanned by two components at the very top in the diagram. If this annulus is essential, ...
3
votes
1answer
193 views

survey paper on the construction of hyperbolic manifolds

Is there a good survey paper which discusses the common ways of building hyperbolic $n$-manifolds?

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