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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Stable normal framings of parallelizable manifolds
Suppose $M$ is a compact, connected, orientable manifold ($\dim M=m$) with trivial tangent bundle and let $j \colon M \to \mathbb R^n$ be an embedding. Suppose we choose a trivialization of $TM$. Then ...
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Component and quasi-component
Let $X$ be a topological space and $x\in X$. Then the quasi-component of the point $x$, denoted by $C_x$, is the intersection of all clopen (closed-and-open) subsets of $X$ which contain the point $x$...
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Are isotopic and conjugate homeomorphisms, conjugate by an element in $\mathrm{Homeo}_0(M)$?
An answer to this question would also answer Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms
Let $M$ be a topological manifold and let $f,g$ be two orientation ...
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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 ...
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Density of closed orbits on hyperbolic surfaces
It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense.
My questions:
If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...
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2
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Behavior of genus function on a 4-manifold for sums
Let $X$ be a smooth compact 4-manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \...
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Representing elements of $\pi_2(M)$ by embedded spheres in 3-manifolds
I am sorry that this question is probably too basic - I could not seem to find the answer though. I know the following - let $S$ be a closed orientable surface, an element of $H_1(S;\mathbb{Z})$ is ...
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Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures
Let $S=S_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x_1,\ldots , x_n\}$ be a set of $n$ distinguished points in the ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 $\frac 1p$ ...
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How difficult is Morse theory on stacks?
The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know.
Suppose I have a Lie ...
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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 $\...
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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 $G$ ...
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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 ...
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Simple proof for property R conjecture
Gabai's property R theorem is:
If the 0-surgery manifold of a knot $K$ is homeomorphic to $S^1\times S^2$, then $K$ is the unknot.
Recently, 3-manifold topology has been developed rapidly by Agol, ...
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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-...
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Topology of boundaries of hyperbolic groups
For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...
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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
$$
H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
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Any 3-manifold can be realized as the boundary of a 4-manifold
We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
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Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?
The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...
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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 ...
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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 ...
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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 $\mathbb E^n$. ...
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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 $\...
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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 ...
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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 ...
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Combinatorial Pin structure
David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the ...
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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 ...
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How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
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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 ...
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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 ...
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The Jones polynomial at specific values of $t$
I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.
...
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Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants
Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is ...
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Essential surfaces in knot complements
Given any knot $K \subset \mathbb{S}^3$, one can find a closed oriented embedded surface $S$ such that $K \subset S \subset \mathbb{S}^3$. Moreover, pick such an $S$ that has minimal genus.
One can ...
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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 ...
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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 ...
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A simple closed curve on a surface
How to describe a simple closed curve on an oriented surface of genus g? I know the answer only for the torus. It would be nice to find an article or a book where proof can be found.
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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?
- 19.2k
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Are there unique geodesics in the NIL and SOL geometry?
Is there a unique geodesics between any two points in the NIL (resp. SOL) geometry?
If so, is there a nice way of parametrizing them? For example geodesics in $S^3$ can be parametrized using the ...
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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 ...
- 40.9k
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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,...
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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, ...
- 10.2k
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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 ...
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