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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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66 views

Integral geometry for general closed smooth manifolds

Let $M$ be a closed smooth manifold of dimension $2n$ for some positive integer $n$. Let $\mathit{Diff}(M)$ be the group of diffeomorphisms of $M$. Let $L$ be a closed embedded $n$-dimensional ...
5
votes
0answers
124 views

Generalizing the formula between Wu class and the Steenrod square

I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy $$ Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) . \tag{eq.1}$$ ...
1
vote
0answers
33 views

Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form $$\beta_y(x)=d(x,y)-d(w,y).$$ The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...
5
votes
1answer
276 views

Are framed manifolds cubulatable?

Let's say an $n$-manifold is cubulated if it is glued out of cubes $[0,1]^n$ in a way that looks locally like the standard cubulation of $\mathbb R^n$. For instance, the face $[0,1]^{k-1} \times \{1\} ...
11
votes
0answers
153 views

What is the centralizer of a Coxeter element?

Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element. If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection ...
3
votes
0answers
26 views

Subalgebras and ideals of algebra of Vassiliev invariants

Let $\mathcal{K}$ be the commutative monoid whose elements are (isotopy) equivalence classes of knots with composition under the connected knot sum, and $\mathbb{Z}\mathcal{K}$ be the corresponding ...
4
votes
0answers
95 views

Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post. I had discussed my computation of $$ \Omega_5^{...
7
votes
1answer
181 views

Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
8
votes
3answers
403 views

SL(2, C)-representation of a knot

When studying knot theory I often encounter $SL(2, \mathbb{C})$-representation of knots (of the knot group) or the $SL(2, \mathbb{C})$ character variety of a knot group. But I just don't seem to ...
0
votes
0answers
275 views

Analogs of “Intersection Forms” for 3-manifolds and 5-manifolds [closed]

We recall the intersection form for 4-manifolds $M^4$, $$Q_{M^4}: H^2(M^4;\mathbb Z)\times H^2(M^4;\mathbb Z) \to \mathbb Z$$ defined by $$ Q_{M^4}(a,b) =\langle a\cup b,[M^4]\rangle=\int_{M^4} ab. $...
4
votes
1answer
85 views

Smooth embedding of space forms in the Euclidean space

I was wondering which $S^n/\Gamma$ can be smoothly embedded into $\mathbb R^{n+1}$, where $\Gamma \subset O(n+1)$ is a finite subgroup. To my knowledge, the case $n \le 3$ is known. It has been proved ...
3
votes
0answers
84 views

About topologically mixing functions

Let us consider a sequence of continuous functions $g_{q}:R^2\to R^2$. Let $(A_{q})_{q\geq 1}$ be a sequence of compact sets in $R^2$. Assuming that each function $g_{q}$ is topologically mixing in $...
12
votes
0answers
186 views

Counter example to lifting contractibility of a topological space

I'm looking for a simple example of an open proper continuous map between topological spaces $\varphi:X\to Y$ such that : $Y$ is contractible and locally contractible ; for any $y\in Y$, $\varphi^{-1}...
2
votes
0answers
86 views

Space of non-vanshing sections path-connected?

Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...
8
votes
0answers
150 views

A question about dimension of SL(2,C) character variety of knot group

It is known that if there isn't a closed essential surface in $S^3 \setminus K$, the dimension of $SL(2,\mathbb C)$ character variety is $1$. (In fact, it holds for a general 3-manifold, not only for ...
5
votes
0answers
82 views

Is there any work in topological data analysis on something like “Voronoi complexes”?

Given a finite set $X \subset \mathbb{R}^n$, we can of course construct the corresponding Čech or Vietoris-Rips filtration. At each level of this filtration the scale parameter is fixed and unrelated ...
3
votes
0answers
57 views

Handlesliding a two component, linking number 1 link

Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...
6
votes
4answers
549 views

Homology sphere with $\mathbb{R}^3$ as the universal cover

Question. Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$? I believe the answer is in the positive and I am looking for (precise) references. If not in ...
3
votes
0answers
67 views

What kind of set is this, spanned by two positive definite matrices?

Let $A$ and $B$ be Hermitian positive definite $n\times n$ matrices over $\mathbb C$ or $\mathbb R$. Then for real $k,\ell,$ the matrix $A^kB^\ell A^k$ is well-defined and again Hermitian positive ...
2
votes
0answers
76 views

Self-intersections of words

Could you give me (some) advice(s) how to compare the self-intersection numbers of reduced cyclic words $w = a^{m_1}b^{n_1}\dots a^{m_k}b^{n_k}$ and $wb = a^{m_1}b^{n_1}\dots a^{m_k}b^{n_k+1}$ (where $...
9
votes
3answers
548 views

Reference request for wild 3-manifolds

I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read ...
0
votes
1answer
63 views

Maximum of a sum of Gaussian functions

Consider the function which maps $\mathbb{R}^n$ to $\mathbb{R}$ \begin{align} f(x) = \sum_{i=1}^{n} b_i\phi_i(x) \end{align} where $\phi_i(x) = \exp(-\frac{||x-x_i||_2^2}{2})$ are Gaussian functions ...
3
votes
0answers
121 views

Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
4
votes
1answer
107 views

non-proper parabolic isometries of hyperbolic spaces

In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\...
5
votes
1answer
163 views

Is there a generalized Property P - what can we say about framed link descriptions of $S^3$?

A knot $K$ is said to have Property P if every nontrivial Dehn surgery on $K$ yields a 3-manifold that is not simply connected. It is known that every knot except the unknot has Property P. I am ...
10
votes
2answers
192 views

Is there a known invariant for knotted surfaces defined by skein relations?

Is there a known invariant for knotted surfaces in $\mathbb{R}^4$ (possibly with additional structures, e.g. colored, framed, etc.) which can be defined using skein relations? By skein relations for ...
11
votes
0answers
149 views

Are there exotic twisted doubles of 4-manifolds?

Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...
7
votes
0answers
194 views

Different definitions of Stiefel-Whitney classes

It is quite easy to show that different definitions of the Stiefel-Whitney classes agree by showing that they satisfy the well-known axioms. Nevertheless I have been asking myself wether one can prove ...
10
votes
1answer
476 views

Homeomorphic characterization of the real line? [duplicate]

Let $A$ be a path-connected subset of $\mathbb R^2$ such that the removal of any singleton from $A$ splits $A$ into two open connected components, each of which is path-connected. Is $A$ necessarily ...
5
votes
0answers
61 views

"There exists $e_0(S)$ such that the shortest nonperipheral curve on $(S, x)$ has extremal length at most $e_0$

I was reading the paper by Masur-Minsky (Geometry of the Complex of Curves I: Hyperbolicity) where they show the curve complex $C(S)$ to be $\delta$-hyperbolic. There given a surface $S$ with ...
5
votes
0answers
114 views

Is there any known relationship between sutured contact homology and Legendrian contact homology?

On one hand, Colin-Ghiggini-Honda-Hutchings' construction (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $L$ in a closed contact manifold $(M,\xi)$ via the sutured contact ...
8
votes
1answer
379 views

Critical dimensions D for “smooth manifolds iff triangulable manifolds”

I am aware that at least for lower dimensions, "smooth manifolds iff triangulable manifolds" at least for dimensions below a certain critical dimensions D. My question is that for ...
6
votes
0answers
89 views

Adjoint of the Hodge de Rham star operator under the integral pairing

Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration ...
4
votes
1answer
145 views

When Stone–Čech compactification is totally disconnected

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...
3
votes
0answers
146 views

If the total space of circle bundle over hyperbolic manifold admits Riemannain metric of non-positive sectional curvature?

If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature? I wish to use the result about the question and find Leeb's work 3-...
4
votes
1answer
199 views

Flat solvmanifolds?

I was looking for some reference on solvmanifolds and came up with a paper by A. Morgan tilted "The classification of flat solvmanifolds". I know there is a complete classification of flat manifolds ...
3
votes
1answer
144 views

Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...
10
votes
0answers
250 views

Triangulation of the complex projective plane

In the 1983 paper ``The 9-vertex Projective Plane'' by W. Kuehnel and T.F. Banchoff (The Mathematical Intelligencer Vol 5.) the authors give a 9 vertex triangulation of the complex projective plane, ...
11
votes
2answers
202 views

Minimal area of Seifert surfaces

Let $K$ be a knot smooth knot in a 3-manifold $M$ and fix a metric on $M$. Let $F$ be a orientable surface of genus $g$ with one boundary component. Then we can consider the family of all maps $\...
9
votes
1answer
384 views

An almost complex structure on $S^2\times …\times S^2 / \mathbb{Z_2}$

Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
3
votes
0answers
257 views

Topological approach to create a space between clouds

I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
4
votes
1answer
111 views

Complement of Donaldson divisors in dimension 4

Let $(X,\omega)$ be a symplectic 4-manifold such that $\omega$ has a rational cohomology class. I am interested in Donaldson divisors (surfaces) $D$ in $(X,\omega)$ whose complement is a 1-handle body....
7
votes
1answer
194 views

Action of diffeomorphism group on non-vanishing vector fields

Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...
4
votes
2answers
192 views

Is the *unreduced* Burau representation unitary?

In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...
8
votes
2answers
366 views

Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?

Let $M$ be a manifold and $V$ be an oriented vector bundle. It's well known that if the Euler class of $V$ is non zero, then $V$ can't have a non-vanishing section. The converse is not true, see ...
9
votes
1answer
408 views

A wild embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$

Can one construct an embedding of $\mathbb{S}^1$ into $\mathbb{R}^3$ so that every orthogonal projection onto a two dimensional plane is a unit disc? It is easy to construct an embedding of $\...
6
votes
1answer
168 views

Relationship between induced maps at homotopy groups level for maps $f:S^2\to S^2$

It is a known result that based point maps $f:S^2\to S^2$ are classified by their degree. That is, by the induced map at $\pi_2$-level $f_{*2}:\pi_2(S^2)\to\pi_2(S^2)$ (the subindex $*2$ just mean it ...
4
votes
0answers
154 views

Mapping class group of $\mathbb{S}^3$

If I recall correctly from a lecture I attended the last year we have that $MCG(\mathbb{S^2})\simeq\frac{\mathbb{Z}}{2\mathbb{Z}}$ by Smale in the 60' and $MCG(\mathbb{S^3})\simeq\frac{\mathbb{Z}}{2\...
7
votes
0answers
167 views

Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
5
votes
1answer
165 views

Approximate Homology of a Large Simplicial Complex

I can use software to calculate the Betti numbers $\beta_0,\beta_1,\beta_2,\dots$ of a finite simplicial complex. This is prohibitive for large complexes, built on say > 100,000 nodes. Is there some ...