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
questions
3
votes
0
answers
87
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 ...
9
votes
4
answers
1k
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
0
answers
86
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 ...
9
votes
3
answers
755
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 ...
1
vote
1
answer
354
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
0
answers
142
views
Can bellows make loops?
Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
4
votes
1
answer
251
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
1
answer
304
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 ...
11
votes
2
answers
436
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 ...
14
votes
0
answers
316
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
0
answers
312
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 ...
17
votes
2
answers
1k
views
Homeomorphic characterization of the real line?
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 ...
6
votes
0
answers
80
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
0
answers
375
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 ...
11
votes
1
answer
839
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
For ...
6
votes
0
answers
331
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 ...
6
votes
1
answer
539
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
0
answers
211
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
1
answer
229
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
1
answer
176
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 $...
11
votes
0
answers
514
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, ...
12
votes
2
answers
456
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 $\...
14
votes
1
answer
798
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) ...
4
votes
0
answers
310
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
1
answer
168
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....
9
votes
1
answer
420
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)...
7
votes
2
answers
450
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
2
answers
472
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
1
answer
638
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 $\...
8
votes
2
answers
387
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 means it ...
4
votes
0
answers
186
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
0
answers
272
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 ...
6
votes
1
answer
327
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 ...
8
votes
0
answers
128
views
"Cross-Ratios" for D_n cluster algebra
Both cluster algebras $A_n$ and $D_n$ admit an interpretation in terms of (tagged) triangulations of Riemann surfaces - respectively a Poincaré disk with n+3 punctures on the boundary and a Poincaré ...
21
votes
6
answers
3k
views
Smooth functions on sphere
Let $u$ be a smooth function defined on the unit sphere $S^2$. Assume $u$ has two local maxima, two local minima, and two saddle points (a total of 6 critical points). Does there exist a plane $P$ ...
22
votes
1
answer
604
views
Presenting 3-manifolds by planar graphs
From a planar graph $\Gamma$, equipped with an integer-valued weight function $d:E(\Gamma) \sqcup V(\Gamma) \to \mathbb{Z}$, one can build a $3$-manifold $M_{\Gamma}$ as follows. For each vertex $v$, ...
5
votes
0
answers
147
views
Categorification-like statement in the cobordism group?
Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...
1
vote
1
answer
753
views
Intersection of hyperspheres
Suppose we have $n$ hyperspheres in $\mathbb{R}^m$, $m\geq n$, of centers $x_1,\ldots x_n$, $x_i\neq x_j\,\forall i,j$, and radii $r_1,\ldots ,r_n$. Suppose that, for every $i,j$, the quantities $r_i$,...
6
votes
0
answers
130
views
Pin cobordism v.s. "KO" theory in low or in any dimensions
Fact: The spin cobordism is equivalent to "KO" theory in low dimension if we only consider the 2-torsion.
This is related to a question and an answer supports the claim.
Here we denote the $p$-...
6
votes
0
answers
216
views
Arf-Brown-Kervaire invariant and a surjective map $G \to Pin^-$
We know that the Arf-Brown-Kervaire (abk) invariant is a bordism invariant of
$$
\Omega_2^{Pin^-}(pt)=\mathbb{Z}/(8\mathbb{Z}),
$$
where the $\mathbb{Z}/(8\mathbb{Z})$ is generated by a 2-manifold $M^...
10
votes
1
answer
356
views
Discrete Pin structures
It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of ...
6
votes
0
answers
82
views
Triple data for Pontrjagin dual of the Spin bordism group
It is said that the Pontrjagin dual of the 3-dimensional Spin bordism group of $BG$ for $G$ a finite group,
$$
\text{Hom}(Ω^{spin}_3(BG),\mathbb{R/Z}),
$$
can be expressed by triples of cochains $$(w, ...
5
votes
0
answers
107
views
Induced new structures on Poincare dual manifolds
"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows
Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
6
votes
0
answers
315
views
Elementary questions about Morse-Bott functions
Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...
7
votes
0
answers
1k
views
Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...
9
votes
2
answers
762
views
Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...
2
votes
1
answer
162
views
Structure sets for three dimensional surgery
Is there a treatment in the literature of the structure sets relating simple homotopy equivalences to homeomorphisms in the three dimensional case? I am aware that due to the ...
4
votes
0
answers
180
views
Does there exist a preferred trivialization of a trivial line bundle?
Let $L\to M$ be a topologically trivial complex Hermitian line bundle (over a manifold of dimension three, if this is of any importance). I assume that $L$ admits a trivialization, however, I do not ...
6
votes
2
answers
237
views
Generalization of Bieberbach's second theorem
Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...
6
votes
0
answers
167
views
Cell structures of simply-connected 5-manifolds (classified by Barden's 1965 paper)
In Barden's 1965 paper: Simply-connected five manifolds, Barden gave a complete list of diffeomorphism classes of simply-connected 5-manifolds:
$$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$
...