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).
4,133
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Exponential contraction for the projection on horospheres
A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a ...
4
votes
1
answer
297
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A Geometric Combinatorial/Graph Theory Question
I have a combinatorics problem that seems pretty general - I'd be surprised if the answer is not known. Unfortunately, I can't seem to solve it.
The question concerns the following situation: ...
5
votes
1
answer
305
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Irreducibility of a Heegaard splitting from its Heegaard diagram
Assume that a Heegaard diagram $(\Sigma_g,\{\alpha_1,\ldots,\alpha_g\},\{\beta_1,\ldots,\beta_g\})$, defining a Heegaard splitting of a closed 3-manifold, is given. So, by attaching 3-dimensional 2-...
11
votes
0
answers
234
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Null-homotopies in the space of framed functions on a surface
Let $M$ be a smooth manifold. Morse functions on $M$ are smooth functions $M \to \mathbb{R}$ with only very nice singularities.
Fact: The space of Morse functions on $M$ is not, in general, ...
6
votes
1
answer
268
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Vanishing of $w_2$ for orientable 3-manifolds
Let $M$ be oriented manifold: this happens if and only if $w_1(M)=0$ (the first Stiefel Whitney class, being an element in $H^1(M,\mathbb{Z}_2)$. There is a result that if $M$ is three dimensional ...
17
votes
1
answer
567
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Simply connected slices
Assume $\Omega$ is an open set in $\mathbb R^3$
such that the intersection of $\Omega$ with any horizontal plane is simply connected.
Can you prove that $\Omega$ is simply connected?
(Note that ...
8
votes
1
answer
440
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Loday's characterization and enumeration of faces of associahedra (Stasheff polytopes)
From "The multiple facets of the associahedra" by Loday:
Let us consider the formal power series
$$f(x) = x+a_1 x^2 +a_2 x^3 + \cdots+ a_n x^{n+1} + \cdots$$
and let
$$ g(x) = x+b_1 x^2 + ...
2
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0
answers
85
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Banach density of a sequence of spheres in a virtually nilpotent group
Let $G$ be a finitely-generated group of polynomial growth equipped with the word metric (with respect to a fixed symmetric generating set).
Let
\begin{equation*}
A = \left\{ g \in G: |g| = mn, n \...
4
votes
0
answers
193
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Integral of second fundamental form
Let us have Riemannian manifold $M$ with boundary $N.$ Let $F$ be an immersion, such that $F:N\to M$ and $B$ be a second fundamental form on $N$ relative to $F.$ And let $f$ be a function on $N.$
...
9
votes
1
answer
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Why is Khovanov homology considered a 'categorification'?
I understand that the Euler characteristic of Khovanov homology is the Jones polynomial. But in what sense does this give category theory structure to the Jones polynomial, i.e., what are the objects ...
3
votes
1
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204
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Fixed-point-free action and cohomology of a finite group
I learned that "If $G$ is a finite group acting freely and continuously on $S^n$, the sphere then $G$ has periodic cohomology".
My question is: Are there any other similar theorems relating the free ...
5
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0
answers
162
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On homeomorphisms non-smooth along submanifolds
Suppose $M_1\supset N_1$ and $M_2\supset N_2$ are two couples consisting of a smooth compact connected manifold $M_i$ with a smooth compact sub-manifold $N_i$.
Suppose there is a homeomorphism $\...
8
votes
1
answer
178
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Heegaard splitting of maps between 3-manifolds
Let $M$ and $M'$ be closed oriented connected 3-manfolds and let $f : M \to M'$ be a continuous map. Do there exist Heegaard splittings $M = H_1 \cup H_2$ and $M' = H_1' \cup H_2'$ and a map $f'$ ...
7
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351
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Making diffeomorphism of submanifolds boring
This is probably very well known in surgery theory... I'm looking for a modern reference on the following questions (the only one I know is Browder's "Diffeomorphism of 1-connected manifolds"...
4
votes
1
answer
213
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Embedding spaces and surface knots in high dimensional manifolds
This is a variation of Craig's Knot complement diffeomorphism groups and embedding spaces for a different type of very simple manifold (surfaces which have a 1-relator fundamental group instead of ...
3
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0
answers
208
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A question about so-called "Brunnian Links" or "Brunnian Rings"
Let E(3) be 3-dimensional Euclidean space with its standard topology. Brunnian Rings are subsets of E(3) which are simple closed curves. It is known that an arbitrarily large set S(3) of these ...
19
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Is there a simpler proof of the key lemma in the paper by Hiroshi Iriyeh and Masataka Shibata on the 3D Mahler conjecture?
In this remarkable paper 30 pages are occupied by the proof of the following innocently looking lemma:
Let $K$ be an origin-symmetric convex body in $\mathbb R^3$. There exist three planes through ...
16
votes
2
answers
766
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Klee's trick --- more applications
In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
10
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4
answers
1k
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Complements of Simply Connected Subsets of the Plane
this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...
13
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0
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174
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Is there a Handle Approximation theorem?
The cellular approximation theorem states that given a continuous map between two CW complexes $f : X \to Y$, then $f$ is homotopic to a cellular map - that is some map $f'$ with $f'(X_n) \subset Y_n$ ...
1
vote
1
answer
171
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Pseudo-Anosov Matrices of surfaces (reference request)
Can the the monodromy matrix of the action of a pseudo-Anosov homeomorphism of a surface on it's homology be of the form of a reducible or block matrix?
Any such reference can be helpful. Thanks.
7
votes
1
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1k
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Mapping class group of certain 3-manifolds
Let $\xi : M^3 \to F$ be an orientable circle bundle over a closed orientable surface $F$ of genus $g \geq 2$. I am mostly interested to the case where the bundle $\xi$ is non-trivial. My question is ...
5
votes
1
answer
163
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Goldman Lie algebra of a bordered surface vs. a closed surface?
How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) ...
21
votes
2
answers
602
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Morphism from a surface group to a symmetric group, lifted to the braid group
Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
6
votes
1
answer
219
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Is the completion of a CAT(0) open ball a closed ball?
It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball!
But CAT(0) spaces are ...
1
vote
0
answers
105
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The inverse image of a Noetherian topological space
A topological space $X$ is called Noetherian if
closed subsets satisfy the descending chain condition, equivalently,
the open subsets satisfy the ascending chain
condition.
Let $A$ and $B$ be ...
7
votes
0
answers
290
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What is the core of the cubic 3-fold?
Let $X$ be a compact, smooth, $1$-connected, orientable $6$-manifold with torsion free homology, then (from Wall's classification) $X$ has a (smooth) connect sum decomposition $X \cong X_{0} \#_{k} S^{...
5
votes
2
answers
467
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Finiteness properties of mapping class groups
Question: Is it known if the mapping class groups (of surfaces of finite type) are similar to Gromov-hyperbolic groups in the following senses:
1) Does every finite generating set give us a finite ...
2
votes
1
answer
187
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Classification of pretzel links up to link homotopy using alexander quandle
I am currently reading this paper where the author classifies the pretzel links up to link homotopy using a quasi-trivial quandle $\mathbb{Z}_{k}[t^{\pm 1}]\diagup_{(t-1)^{2}}$, and I find it ...
4
votes
1
answer
256
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Invariance of Khovanov homology under first Reidemester move
I am studying Khovanov homology from five lectures on Khovanov homology
and I want to try to show Khovanov homology is invariant under first Reidemester move but I cannot understand how we can write
...
5
votes
2
answers
237
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Monoid of continuous self-maps of (real) surfaces
Let $S$ be a closed surface of genus $g > 0$ and $[S,S] = Hom(\pi_{1}(S),\pi_{1}(S))$ be the monoid of (homotopy classes of) continuous maps from $S$ to itself. Consider the semi-group $A$ of ...
6
votes
1
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250
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Simplicial approximation of a fibration
I am interested in the simplicial approximation of Serre or Hurewicz fibrations (or even fibre bundles). Let's assume $E$ and $B$ are finite simplicial complexes (or their associated geometric ...
3
votes
1
answer
135
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Submanifold generator of Pontryagin-dual of the torsion subgroup of the $Pin^+$ bordism group (dimension 4th)
I am interested in figuring out the following submanifold generator of a $Pin^+$ Bordism or Cobordism group in the dimension 4, say for a $Pin^+$ cobordism group of the classifying space $BG=BSU(2)$:
...
3
votes
1
answer
498
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Linking number a complete invariant of link homotopy
I read in Milnor's article "Link groups", where he defines invariants to classify links up to link homotopy, that the linking number is a complete invariant which can tell almost trivial two ...
11
votes
2
answers
930
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Subtle point in definition of BNS invariant
Let $G$ be a finitely generated group. Let $S(G)$ be the quotient of $\text{Hom}(G,\mathbb{R}) \setminus \{0\}$ by the equivalence relation that identifies two homomorphisms if they differ by scaling ...
4
votes
1
answer
644
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Lifting cellular structures to fibrations, fibre bundles or coverings
It is a well known result in Algebraic Topology that given a covering space $E\to B$ where the base has a CW-structure, then the total space can be given a CW-structure (see for example Theorem 8.10 ...
3
votes
0
answers
99
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Multiple fibrations of a 3-manifold over the circle [duplicate]
I am looking for the simplest explicit example of a $3$-dimensional manifold $M$ which is a surface fiber bundle over the circle $S^1$ in two different ways -- with fibers $F_1$ and $F_2$ respectively....
8
votes
1
answer
648
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Surgery along an embedded surface in a 4-manifold
Let $X$ be a 4-manifold and $\Sigma_g\subset X$ be an embedded closed orientable genus = $g$ surface. Suppose $\Sigma_g\subset X$ has a trivial closed normal bundle $N(\Sigma_g) = \Sigma_g\times D^2$. ...
5
votes
1
answer
190
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Equivariant cohomology defined by restrictions?
Suppose that $G=S^1$ acts on a smooth, connected, compact manifold with discrete fixed points, additionally assume that there is at least one fixed point.
Let $\alpha \in H^{2}_{S^1}(M)$ be such ...
11
votes
2
answers
1k
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Why is $\mathbb{Z}$ not a Kähler group?
Is there some simple proof that $\mathbb{Z}$ is not isomorphic to the fundamental group of any compact Kähler manifold? This follows from the main result of https://arxiv.org/abs/0709.4350 which ...
10
votes
2
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683
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Representability of the sum of homology classes
This is probably a very simple question, but I have not found it addressed in the references that I know. Let $M$ be a closed and connected orientable $d$-dimensional manifold ($d\leq 8$) and let $[\...
7
votes
2
answers
587
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Which topological spaces contain dense simply connected subspace?
And when can this subspace be chosen to be open?
As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds ...
5
votes
0
answers
235
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Fibered 3-manifolds over circle with harmonic projection map
Suppose we have a surface bundle $M$ over the unit circle $S^1$ and $M$ is assigned with Riemannian metric $g$. The projection map $\pi: M\to S^1$ now can be homotopic to a harmonic map $\phi: M\to S^...
8
votes
1
answer
345
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1-Bridge Braids in Solid Tori (Berge-Gabai knots), dual knots, & knot exteriors
Let $K$ be a knot in a solid torus. Combining results of Berge and Gabai, we know that if $K$ admits a solid torus surgery, then $K$ is a 1-bridge braid. Using Gabai's result, we can figure out what ...
21
votes
2
answers
806
views
Do Betti numbers beyond the first have a "number of cuts" interpretation?
I have heard stated the following
Theorem. If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you ...
2
votes
0
answers
135
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A gradient trajectory connecting boundary components in an annulus
In a course of writing a paper I realised that I need a lemma below, and found a half page proof. I have not seen this lemma before and wonder if maybe someone here knows this lemma or can prove it in ...
6
votes
1
answer
600
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Proof of the stable homeomorphism conjecture
I would like to get a demonstration of the stable homeomorphism conjecture (SHC$_n$), stating that any orientation-preserving homeomorphism $\mathbb{R}^n \rightarrow \mathbb{R}^n$ is stable in ...
3
votes
0
answers
149
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Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
12
votes
3
answers
949
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Fixed point set of smooth circle action
Suppose $M$ is a connected closed smooth $d$-dimensional manifold, and suppose $S^1 = SO(2)$ acts smoothly on $M$. Then the fixed point set $Y = M^{S^1}$ will be a submanifold of $M$ of even ...
12
votes
2
answers
677
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Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?
Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \...