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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,...

3
votes
0answers
225 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 ...
3
votes
1answer
87 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....
6
votes
1answer
178 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
174 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 $s^2 = t$ ...
7
votes
2answers
281 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 ...
8
votes
1answer
345 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
148 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
137 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\...
6
votes
0answers
140 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
157 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
0answers
69 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
7answers
2k 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 $P$ ...
14
votes
0answers
183 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
0answers
121 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 $$ \Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots, $$ ...
0
votes
1answer
111 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
0answers
100 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
0answers
138 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^...
9
votes
1answer
133 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 ...
5
votes
0answers
56 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
0answers
83 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 $\text{Pin}^-$...
5
votes
0answers
270 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 ...
5
votes
0answers
222 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 \\ ...
8
votes
2answers
288 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
1answer
142 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
0answers
138 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
2answers
169 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
0answers
112 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: $$X_{j,k_1,\dots,k_n}=X_j\#M_{k_1}\#\cdots\#M_{k_n}$$ ...
4
votes
1answer
168 views

A combinatorial gadget for smoothly slice knots

As far as I searched, I couldn't find something valuable but is there any combinatorial (or computable) gadget for knots to guarantee them to be smoothly slice? For example, by the virtue of the ...
6
votes
2answers
287 views

A counter-example for the reversed direction of Casson-Gordon's theorem

For a knot $K$, let $\Sigma_K$ be the double cyclic branched cover of a knot. By the classical work of Casson and Gordon, we know that if $K$ is smoothly slice, then $\Sigma_K$ bounds a rational ...
2
votes
0answers
74 views

Simple homotopy type of interval bundles over surfaces

Consider a locally trivial (topological) bundle over the Klein bottle $$ I\to E \to K$$ The projection map $E \to K$ is a homotopy equivalence. Is it a simple homotopy equivalence? ...
4
votes
0answers
67 views

Linear Independence & Integral Homology Cobordism Group

The set of integral homology spheres up to integral homology cobordism forms an abelian group with the operation induced by the connected sum. This group is called integral homology cobordism group ...
3
votes
0answers
61 views

Syzygies in Steinberg Module of genus 2 mapping class group $MCG(\Sigma_2)$

Consider the mapping class group $MCG(\Sigma_2)$ of the closed genus 2 oriented surface $\Sigma_2$. The algebraic-duality theory of $MCG_2:=MCG(\Sigma_2)$ is explicitly described by Nathan Broaddus' ...
5
votes
0answers
155 views

Homology spheres bounding homology balls but not embedding into $S^4$

Are there any examples of integer homology spheres $Y^3$ that bound smooth integer homology balls but that do not smoothly embeded into $S^4$?
5
votes
1answer
186 views

smooth homotopy 4-balls with sphere boundary in dimension 4

What follows is, as far as I can tell, totally standard folklore. I have one particular point of confusion, other than that, I wanted to confirm that I am uttering the incantations correctly. The ...
4
votes
0answers
91 views

Bound on critical points of Lefschetz fibration over the disk with prescribed monodromy

Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
2
votes
0answers
117 views

Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data: Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$ in a 2 dimensional (2d) spacetime. ...
8
votes
0answers
100 views

Relating bordism generators in d and d+2 dimensions — an explicit example

This is an attempt to make my relation between bordism invariants in $d$ and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
2
votes
1answer
124 views

Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group

Let $M^d$ be a nontrivial bordism generator for the bordism group $$ \Omega_d^G= \mathbb{Z}_n, $$ suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
8
votes
0answers
139 views

“Gerbes” in the cobordism theory

In a lecture I attended today, I heard the use of gerbes in the cobordism theory. Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...
4
votes
0answers
64 views

Relating bordism invairants in $d$ and $d+2$ dimensions

Are there some relationship between mapping the bordism invairants of eq.1 and eq.2? $$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$ and $$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...
6
votes
1answer
153 views

Framings for 2-surgeries on 4-manifolds

I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$. Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible ...
5
votes
1answer
188 views

Manifold generators of O-bordism invariants

If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...
1
vote
0answers
30 views

Defining a notion of “volume of its lattice” for non-rational subspaces

Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\...
6
votes
1answer
105 views

Example of nonvanishing Waldhausen Nil group

In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen ...
6
votes
2answers
250 views

Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
30
votes
2answers
540 views

Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
2
votes
0answers
54 views

Topological Shape Operator More Sensitive than Inverse Limits

This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
4
votes
0answers
83 views

Simple invariants to detect concordance in general 3-manifolds

Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that ...
8
votes
1answer
193 views

Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

$\require{AMScd}$ Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this. Theorem 1: (Lagrangian Neighborhood Theorem) ...
6
votes
0answers
65 views

Stable commutator lengths of pseudo-Anosovs

Does anyone have an example of a pseudo-Anosov mapping class for which the stable commutator length is known exactly?