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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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 ...
M. Dus's user avatar
  • 1,900
4 votes
1 answer
297 views

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: ...
John Samples's user avatar
5 votes
1 answer
305 views

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-...
Mustafa's user avatar
  • 349
11 votes
0 answers
234 views

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, ...
Adam Saltz's user avatar
6 votes
1 answer
268 views

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 ...
truebaran's user avatar
  • 9,140
17 votes
1 answer
567 views

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 ...
Anton Petrunin's user avatar
8 votes
1 answer
440 views

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 + ...
Tom Copeland's user avatar
  • 9,937
2 votes
0 answers
85 views

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 \...
user94744's user avatar
4 votes
0 answers
193 views

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 $N.$ ...
Dozzy Cooper's user avatar
9 votes
1 answer
1k views

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 ...
Mtheorist's user avatar
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3 votes
1 answer
204 views

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 ...
user60933's user avatar
  • 481
5 votes
0 answers
162 views

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 $\...
aglearner's user avatar
  • 14k
8 votes
1 answer
178 views

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 $f'$ ...
user101010's user avatar
  • 5,319
7 votes
0 answers
351 views

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"...
Nati's user avatar
  • 1,971
4 votes
1 answer
213 views

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 ...
Nati's user avatar
  • 1,971
3 votes
0 answers
208 views

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 ...
Garabed Gulbenkian's user avatar
19 votes
0 answers
621 views

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 ...
fedja's user avatar
  • 59.5k
16 votes
2 answers
766 views

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 $K$ ...
Anton Petrunin's user avatar
10 votes
4 answers
1k views

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 \...
John Samples's user avatar
13 votes
0 answers
174 views

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 $f'(X_n) \subset Y_n$ ...
user101010's user avatar
  • 5,319
1 vote
1 answer
171 views

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.
hakunamatata's user avatar
7 votes
1 answer
1k views

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 ...
Daniele Zuddas's user avatar
5 votes
1 answer
163 views

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) ...
Nati's user avatar
  • 1,971
21 votes
2 answers
602 views

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\...
Gael Meigniez's user avatar
6 votes
1 answer
219 views

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 ...
Brian Rushton's user avatar
1 vote
0 answers
105 views

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 ...
Zerolex's user avatar
  • 11
7 votes
0 answers
290 views

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^{...
Nick L's user avatar
  • 6,923
5 votes
2 answers
467 views

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 ...
Mehdi Yazdi's user avatar
2 votes
1 answer
187 views

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 ...
Suki's user avatar
  • 55
4 votes
1 answer
256 views

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 ...
bytrz's user avatar
  • 141
5 votes
2 answers
237 views

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 ...
Nick L's user avatar
  • 6,923
6 votes
1 answer
250 views

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 ...
D1811994's user avatar
  • 909
3 votes
1 answer
135 views

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 $BG=BSU(2)$: ...
annie marie cœur's user avatar
3 votes
1 answer
498 views

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 ...
Suki's user avatar
  • 55
11 votes
2 answers
930 views

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 ...
Andy Putman's user avatar
  • 43.4k
4 votes
1 answer
644 views

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 ...
D1811994's user avatar
  • 909
3 votes
0 answers
99 views

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....
Adam's user avatar
  • 2,370
8 votes
1 answer
648 views

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 $N(\Sigma_g) = \Sigma_g\times D^2$. ...
Shinichiro Nakamura's user avatar
5 votes
1 answer
190 views

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 ...
Nick L's user avatar
  • 6,923
11 votes
2 answers
1k views

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 ...
Nick L's user avatar
  • 6,923
10 votes
2 answers
683 views

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 $[\...
Bilateral's user avatar
  • 3,064
7 votes
2 answers
587 views

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 ...
erz's user avatar
  • 5,385
5 votes
0 answers
235 views

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^...
Donghao's user avatar
  • 161
8 votes
1 answer
345 views

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 ...
Krishna's user avatar
  • 561
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 ...
Qfwfq's user avatar
  • 22.7k
2 votes
0 answers
135 views

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 ...
aglearner's user avatar
  • 14k
6 votes
1 answer
600 views

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 ...
N. Roobaert's user avatar
3 votes
0 answers
149 views

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 ...
Guangbo Xu's user avatar
  • 1,197
12 votes
3 answers
949 views

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 ...
Jens Reinhold's user avatar
12 votes
2 answers
677 views

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 \...
Pablo's user avatar
  • 11.2k

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