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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,...

3
votes
3answers
163 views

Extending a continuous map over projective space

Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
1
vote
0answers
63 views

Universal cover of ladderly puntured complex plane

The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function. I am looking for the constructions of the covering map from the ...
6
votes
1answer
117 views

Reference request: Can iterated torus links be mutated?

I believe that most iterated torus links cannot be changed non-trivially by a Conway mutation, as follows. If you look at the JSJ decomposition of the double-branched cover, then each satellite torus ...
1
vote
0answers
74 views

Determine all possible magnetic monopole of gauge theories

In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact: This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...
10
votes
0answers
116 views

A geometric interpretation of the odd-primary Kervaire elements

Let $\Omega^\mathrm{fr}_\ast \cong \pi_\ast S$ denote the graded ring of cobordism classes of framed manifolds, which, by the Pontryagin-Thom construction, is isomorphic (as a graded ring) to the ...
2
votes
0answers
40 views

Must bounded, closed, smooth curves with long straights have sharp bends?

Consider the family of bounded, closed, and continous curves $\Gamma$, i.e. for all $\gamma \in \Gamma$, we have $\gamma : [0, 1) \to [0, 1]^2$. Within this family, I am interested in curves that ...
10
votes
2answers
327 views

Is the boundary of a manifold topologically unique? [duplicate]

Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \...
9
votes
0answers
138 views

Generalize $\mathbb Z/p$-space for irrational $\alpha$

A free $\mathbb Z/p$-space is a topological space $X$ with an action $\varphi$ such that $\forall x\in X$ $\varphi^p(x)=x$ but $\varphi(x)\ne x$. I would like to generalize this notion from $\frac 1p$ ...
7
votes
1answer
256 views

Higher dimensional scutoids?

The recent discovery of scutoids in biological structures is fascinating. Two scutoids are depicted below (from Scientists Have Discovered an Entirely New Shape, And It Was Hiding in Your Cells), each ...
5
votes
1answer
139 views

Discrete approximations of Riemannian manifolds

MSE crosspost It's known (due to Perelman) that in class of Alexandrov spaces of fixed dimension and bounded from below curvature Gromov-Hausdorff distance separates homeomorphism types — every $\...
7
votes
4answers
244 views

Lattices of PU(n,1) with large abelianization

I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...
4
votes
2answers
338 views

Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$

I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}...
6
votes
0answers
145 views

What is the mathematical structure of 2d TQFT from the 2d foam category (instead of 2d cobordism category)?

It is well-known that the category of 2d TQFTs is equivalent to the category of commutative Frobenius algebras. What about functors from the 2d foam category (instead of 2d cobordism category) to ...
1
vote
1answer
78 views

framed n component link on a genus n surface determines the self-homeomorphism?

$S$ is the boundary of a genus $n$ handlebody in $S^3$. $\{m_1, m_2,..., m_n\}$ is the collection of the meridian circles of $S$; $\{l_1,l_2,...,l_n\}$ is the collection of the longitude circles on $S$...
1
vote
0answers
35 views

From Sudakov minoration principle to lowerbounds on Rademacher complexity

For a compact subset $S \subset \mathbb{R}^n$ (and an implicit metric $d$ on it) and $\epsilon >0$ lets define the following $2$ standard quantities, Let ${\cal P}(\epsilon,S,d)$ be the $\epsilon-...
11
votes
1answer
279 views

Obstruction of spin-c structure and the generalized Wu manifods

Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the $$ H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
7
votes
1answer
208 views

Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold. Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
5
votes
2answers
400 views

Any 3-manifold can be realized as the boundary of a 4-manifold

We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
9
votes
0answers
129 views

Define the 3d Chern-Simons TQFT on a discrete simplicial complex

Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...
10
votes
2answers
183 views

Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?

The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$, $$\mathrm{sl}(K) \le - \chi(\Sigma)$$ for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...
16
votes
2answers
412 views

What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?

The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
7
votes
1answer
296 views

Random 3-manifolds in $R^4$

Consider following program: Generate random 3-manifold embedded in $R^4$. Perform its triangulation. Put it to Regina and calculate what manifold it is. Assuming that we have good algorithm for ...
26
votes
1answer
412 views

What is the minimal dimension of a complex realising a group representation?

This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex). Many interesting integral ...
9
votes
1answer
247 views

Finite group representation as $\mathrm{Aut}(\Gamma)$ action $H^1(\Gamma,\mathbb{Z})$ of graph?

Let $\Gamma$ be a finite graph, then $H^1(\Gamma,\mathbb{Z})\cong \mathbb{Z}^{g(\Gamma)}$ can be viewed as a $\mathrm{Aut}(\Gamma)$ module. Conversely, given a finite group $G$, and a $G$-module $\...
6
votes
1answer
235 views

Generalized projective spaces, spheres, and exotic spheres [closed]

I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres: The real projective space $\mathbb{RP}^1 \simeq S^1,$ is ...
4
votes
0answers
144 views

Vanishing cycles for elliptic fibration on K3 surface?

Let $X$ be an elliptic K3 surface (over $\mathbb{C}$). Assume we have an elliptic fibration on $X$ that only has $I_1$ singular fibers. If we fix a smooth fiber $F$ of such a fibration and a ...
5
votes
0answers
165 views

Asphericity of hypersurface complement in ${\mathbb C}^n$

How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$. One way I can think of is to give ...
4
votes
1answer
124 views

Minimum dimension of faithful representation of mapping class groups?

Let $\Sigma_{g}$ be a closed orientable surface of genus $g$. Let $d_g$ denote the minimum dimension of a faithful representation of the mapping class group of $\Sigma_g$. For $g=1$, the mapping class ...
31
votes
3answers
2k views

Why should I care about the Jones polynomial?

The invention of the Jones polynomial led to hundreds of papers and a Fields medal. However, as far as I can tell it had few consequences in topology. After all, after Thurston’s work we already ...
2
votes
1answer
272 views

Non-orientable 3-manifolds

I am reading "Non-orientable 3-manifolds of small complexity" by Amendola, Martinelli. In this work $\mathbb P^2$-irreducible complexity 6 manifolds are listed. There are 5 of them. I wonder about the ...
6
votes
1answer
184 views

What is the complexity of determining if a knot group is $\mathbb{Z}$?

It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then: The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...
10
votes
0answers
211 views

Amalgamated product of automatic groups

In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic? Is this ...
5
votes
1answer
159 views

How many simple closed geodesics in a given primitive homology class?

It is well-known that an essential closed curve on a hyperbolic surface (possibly with boundary) is homotopic to a unique closed geodesic. Moreover, if the curve under consideration is simple, then so ...
8
votes
1answer
184 views

Bi-Lipschitz extension

Given a bi-Lipschitz homeomorphism $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
3
votes
1answer
154 views

Isotopy Invariants of 2-manifold

What are the ambient isotopy invariants of a 2-manifold with boundary embedded in $R^3$? Is there a good reference for the case of genus 0?
6
votes
1answer
134 views

Can one construct a regular neighborhood without an ambient space?

If I understand my PL topology correctly (and please correct me if I don't), if $K$ is a $k-$complex and $n\ge 2k+2$, then any two PL embeddings $a,b\colon K\to \mathbb{R}^n$ are isotopic. Therefore, ...
2
votes
1answer
217 views

What are some surprising facts that happen after you remove a point to a space? [closed]

There are some facts that are really impressive after you remove a point to a space. Some typical examples are the existence of exotic spheres or the fact that $S^4$ is not almost complex. Or some not ...
5
votes
1answer
221 views

Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action

Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...
16
votes
1answer
333 views

Lowest Dimension for Counterexample in Topological Manifold Factorization

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...
6
votes
1answer
158 views

Five-dimensional manifolds fibering over a fixed hyperbolic surface

I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
9
votes
1answer
131 views

Projective resolutions of finite-dimensional representations of infinite groups

Let $G$ be a group and let $V$ be a finite-dimensional complex representation of $G$. Question: Under what circumstances can I find a projective resolution $$ \cdots \longrightarrow P_3 \...
27
votes
1answer
618 views

Four circles on the sphere

Consider configurations consisting of 4 distinct circles on the sphere. Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to enumerate/...
4
votes
0answers
88 views

Gaussian curvature/Euler characteristic of Facebook clusters

If I look at a connected subgraph on a small collection of actors (such as a small cluster) in the Facebook social network, and I find that 1) The Euler characteristic of the clique complex built on ...
5
votes
1answer
284 views

Formula for Goldman Lie bracket of surface

Let $\Sigma_{g}$ be a closed oriented surface of genus $g$, Goldman defined a Lie algebra structure on the free module generated by the free homotopy classes of loops on $\Sigma_{g}$. Roughly speaking,...
3
votes
0answers
146 views

Step by step construction of 3-manifolds in $R^4$

My question has survived. Therefore I try another one. Consider some elementary operations on closed compact 3-manifold $M \subset R^4$. These elementary operations are e.g. $0$-surgery or $1$-surgery ...
3
votes
1answer
128 views

Links defined by link-severance tableau

Consider a finite $n$-element classical (real) link and the resulting link structure obtained by cutting each of the component elements (knots). Let us represent the resulting structures in a tableau,...
4
votes
1answer
160 views

Are fundamental groups of web complements residually finite?

While thinking of whether any web (spatial trivalent graph) without an embedded bridge can be realized as a branching locus of a finite branched cover over $S^3$, I realized that this problem is ...
7
votes
0answers
124 views

Smale's relative h-cobordism theorem

In Smale's On the structure of manifolds paper there is his relative version of the h-cobordism theorem, specifically Theorem 3.1 (and 1.4). Roughly speaking this concerns the situation where one has ...
3
votes
1answer
108 views

Isotopy extension theorem: how non-unique is ambient isotopy

Let $M$ and $N$ be smooth manifolds. Consider an isotopy of $M$ inside $N$. This means that we have a level preserving embedding $J\colon M\times [0,1] \to N \times [0,1]$. Put $J(x,t)=(\phi_t(x),t)$. ...
8
votes
0answers
112 views

Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms

Let $\Sigma$ be an oriented compact surface with non-empty boundary that is not a disk or a cylinder (i.e. negative Euler characteristic). Let $\phi, \psi: \Sigma \to \Sigma$ be two orientation ...