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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The James and Morse filtrations of homotopy groups

Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
Tyrone's user avatar
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4 votes
0 answers
117 views

Triangulating piecewise-linear manifolds

Question 1: Is this the mainstream definition of a PL-manifold? Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
Vadim's user avatar
  • 346
2 votes
1 answer
310 views

Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries

By Bers' simultaneous uniformization theorem, if $\Gamma$ is a Fuchsian group, then $\operatorname{QC}(\Gamma)\cong \mathcal{T}(S)\times\mathcal{T}(\overline{S})$ where $S = \Bbb H^2/\Gamma$. In ...
one potato two potato's user avatar
6 votes
1 answer
264 views

Generating 21-vertex 4-regular plane graphs with 8 faces of degree 3 and 15 faces of degree 4

Is there any way to generate all 4-regular plane graphs with 21 vertices, 8 faces of degree 3, and 15 faces of degree 4? If so, how many of these graphs are there and what are they?
Xin Zhang's user avatar
  • 1,130
0 votes
1 answer
335 views

Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]

We know that framing structure means the trivialization of tangent bundle of manifold $M$. string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
zeta's user avatar
  • 337
3 votes
0 answers
217 views

What is known about the map $\text{Mod}_g^1 \rightarrow \text{Aut}(F_{2g})$?

Follow up question, edited in on 12/20 below: Letting $\text{Mod}_g^1$ be the mapping class group of a surface with one boundary component (and basepoint on the boundary) and identify its fundamental ...
Chase's user avatar
  • 49
5 votes
1 answer
331 views

Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$ I have a reference request for a proof for the following statement in the title: The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
Chaitanya Tappu's user avatar
3 votes
0 answers
210 views

Classifying spaces beyond CW complexes

We know that for a reasonable topological group $G$ (say a compact Lie group) admits a classifying space for $G$-bundles within the category of countable CW complexes. That means, there is a space $BG$...
UVIR's user avatar
  • 971
4 votes
0 answers
111 views

Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
Sagnik Biswas's user avatar
4 votes
1 answer
134 views

Salvetti complex of dihedral Artin group

The Salvetti complex of a RAAG is well-known and it is fairly simple, since each complete graph gives rise to a tori. The case of Artin groups is wilder, since we do not have tori anymore. The ...
Marcos's user avatar
  • 577
0 votes
1 answer
112 views

Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds

In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as $$g = \frac{...
JMK's user avatar
  • 301
2 votes
0 answers
105 views

Extension of isotopies

In what follows $M$ will be a manifold (without boundary, for simplicity) and $C\subseteq M$ will denote a compact subset. In the paper Deformations of spaces of imbeddings Edwards and Kirby prove the ...
Tommaso Rossi's user avatar
4 votes
2 answers
530 views

Compactification of a product of manifolds

Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
zarathustra's user avatar
1 vote
1 answer
135 views

Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms? Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
zeta's user avatar
  • 337
7 votes
1 answer
424 views

Groups acting on infinite dimensional CAT(0) cube complex

I have seen many examples where a finitely generated infinite group acts properly/freely by isometry on finite dimensional CAT(0) cube complexes. Examples of such groups are discussed in many articles....
bishop1989's user avatar
1 vote
0 answers
141 views

Minimal first Pontryagin class $p_1=1$?

From Hirzbuch theorem, the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$. I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$. Is ...
zeta's user avatar
  • 337
3 votes
0 answers
121 views

Stochastic braids

I am definitely not a probability guy, but I'd like to have a heuristic answer to the following question: do $n$ independently moving points in an open, connected, bounded region $R$ tend to "...
Andrea Marino's user avatar
9 votes
2 answers
569 views

Generalization of the sphere theorem in dimension at least 4

In 1956, Papakyriakopoulos proved Dehn's lemma, loop theorem and the sphere theorem. The proofs are based on a clever technique called "tower construction". Later, Whitehead, Shaprio, ...
Shijie Gu's user avatar
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4 votes
1 answer
293 views

Induced fiber sequence and Eilenberg–MacLane space in Whitehead tower of $BO$

In Whitehead tower of $BO$, there is a induced fiber sequence: 1. $$ Z_2 \to B SO \to BO \overset{w_1}{\rightarrow} B Z_2 $$ How does this map $\overset{w_1}{\rightarrow}$ from $BO$ to $B Z_2$? ...
zeta's user avatar
  • 337
0 votes
0 answers
67 views

Topological transversality by dimension

We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
UVIR's user avatar
  • 971
2 votes
1 answer
142 views

string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$

Why do the string bordism group and the framed bordism group coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)? Why do the string bordism group and the framed bordism group differ ...
wonderich's user avatar
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1 vote
0 answers
125 views

Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?

What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower? Namely, how do we know $$ K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)? $$ Naively -- in each step ...
zeta's user avatar
  • 337
3 votes
0 answers
111 views

Universal cover of the configuration space of points on surface

Let $S$ be a closed oriented surface and $C(S, n)$ be the configuration space of $n$ points on $S$, i.e., the space of $n$-tuples of distinct points of $S$ with the topology induced from $S^n$. Let $V ...
Roman's user avatar
  • 173
0 votes
2 answers
310 views

If a graph embedded on a surface is divided by a curve into a right and left that do not intersect can it be embedded on a surface of smaller genus?

Suppose we have a graph $G$ embedded on a (smooth, orientable etc) surface $Q$. Suppose there is a cycle $C$ of $G$ such that $C$ does not separate our surface $Q$ into two connected regions and ...
Hao S's user avatar
  • 161
4 votes
1 answer
260 views

Casson's knot invariant

$\DeclareMathOperator\SU{SU}$Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology $3$-sphere $M$ into the ...
Partha's user avatar
  • 759
4 votes
1 answer
420 views

Detecting a "bad map" in Fintushel-Stern knot surgery

Background Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...
rab's user avatar
  • 139
1 vote
0 answers
121 views

Can a closed null-homotopic curve be filled in by a disc?

Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
M. Winter's user avatar
  • 12.5k
7 votes
1 answer
243 views

Does the continuous image of a disc contain an embedded disc?

Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
M. Winter's user avatar
  • 12.5k
7 votes
2 answers
436 views

Amalgamated product acting on CAT(0) cube complex

I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger. Result: Let $F_0,F_1$ and $H$ be groups acting properly by isometries on complete $...
bishop1989's user avatar
1 vote
0 answers
138 views

Uniqueness of geometric simplicial structure on $\mathbb{R}^n$

By work of Stallings we know that $\mathbb{R}^n$ (for $n\neq4$) has unique simplicial structure up to equivalence. If instead of a general simplicial structure we consider only geometric simplicial ...
Prateep's user avatar
  • 141
5 votes
1 answer
184 views

Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic space

Let $\Gamma$ be a hyperbolic group. Let $g$, $\gamma\in \Gamma$ freely generate a non-abelian semigroup (in particular, they don't commute and have infinite order). Does the equation $g\gamma^n=h^m$ ...
Yanlong Hao's user avatar
4 votes
1 answer
167 views

Version of pseudo-isotopy $\neq$ isotopy for $(n+1)$-framings

Let $M$ be a closed $n$-manifold and $\varphi$ be a self-diffeomorphisms of $M$. There is a bordism from $M$ to itself given by $M\times [0,1]$ with the identification $M \cong M \times \{0\}$ induced ...
Daniel Bruegmann's user avatar
16 votes
0 answers
390 views

Is the oriented bordism ring generated by homogeneous spaces?

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
Zhenhua Liu's user avatar
7 votes
1 answer
112 views

Picture of the isotopy class of a degree $d$ smooth complex curve

All smooth complex curves of degree $d$ in $\mathbb{C}P^2$ are isotopic. Let $C$ be such a curve. I often picture $\mathbb{C}P^2$ as a 2-dimensional disk bundle over $S^2$ (of Euler class 1) which ...
Sprotte's user avatar
  • 1,045
6 votes
0 answers
199 views

"Inclusion" between higher categories of framed bordisms?

Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds. It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences. If $n$ is large enough, ...
Daniel Bruegmann's user avatar
1 vote
2 answers
221 views

Link invariants from Hecke relations of higher order

Alexander theorem says oriented links in $\mathbb{R}^3$ can be represented by closures of braids. Markov theorem says that braids related by Markov moves produce isotopic braid closures, and vice ...
Student's user avatar
  • 5,008
2 votes
1 answer
212 views

Least number of ways to color a map using at most 4 colors

Given all the maps with k regions, consider the map(s) that can be colored in the least amount of possible ways. In how many ways can it be colored (n)? (using at most 4 colors and not counting ...
Dr.X's user avatar
  • 81
5 votes
1 answer
273 views

Kirby diagrams of Mazur manifolds

In the 1980's, Fintushel-Stern and Fickle independently proved that Brieskorn spheres $\Sigma(2,3,25)$ and $\Sigma(3,5,19)$ bound some Mazur type contractible 4-manifolds with a single $0$-, $1$, and $...
Upside Down's user avatar
2 votes
0 answers
128 views

Need help understanding the geometry of a particular building structure

$\DeclareMathOperator\SL{SL}$I’m not primarily a geometer, so apologies if this question is worded poorly. I’ve been looking at asymptotic cones of connected semisimple Lie groups with at least one ...
jsch's user avatar
  • 21
4 votes
1 answer
221 views

Borel cohomology for circle actions on odd spheres

Suppose we have a $S^1$-action on the odd sphere $S^3$ as follows: $$ \lambda \cdot (z_1, z_2) = (\lambda \cdot z_1, \lambda^2 . z_2)$$ I would like to understand the (Borel) equivariant cohomology of ...
Aditya De Saha's user avatar
1 vote
1 answer
197 views

Moving of sphere embedding and its interior defined by Jordan-Brouwer separation theorem

Let $f_1:\mathbb S^{n-1}\rightarrow \mathbb R^n$ be a continuous embedding, where $\mathbb S^{n-1}$ is the unit sphere of dimension $n-1$, and a point $x$ in the interior of $f_1(\mathbb S^{n-1})$ ...
Tian LAN's user avatar
  • 135
2 votes
2 answers
235 views

Is a simple closed curve always a free boundary arc?

Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points? For a simple closed curve $\...
Pacific saury's user avatar
18 votes
1 answer
2k views

Intuition behind manifolds which are homeomorphic but not diffeomorphic

Popular articles on mathematics often explain the difference between homeomorphism and diffeomorphism with statements like - "A rectangle is homeomorphic to the circle but not diffeomorphic to it&...
Anindya's user avatar
  • 371
5 votes
1 answer
369 views

Branched coverings of non-orientable 3-manifolds

A continuous map of 3-dimensional manifolds $f \colon M^3 \to N^3$ is called a branched covering if there is a link $L \subset N^3$, such that the restriction $f \colon M \setminus f^{-1}(L) \to N \...
vladimir smurygin's user avatar
10 votes
1 answer
555 views

If the universal cover has three boundary components, does it have infinitely many?

Suppose that $M$ is a compact, connected three-manifold with boundary. Suppose that $\pi_1(M)$ is infinite. Suppose that $\tilde{M}$, the universal cover of $M$, has at least three boundary components....
Sam Nead's user avatar
  • 25.9k
5 votes
0 answers
236 views

Aspherical space whose fundamental group is subgroup of the Euclidean isometry group

Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
Chicken feed's user avatar
2 votes
0 answers
81 views

The space of immersions of a loop in a surface

Let $\Sigma$ be a compact oriented surface with boundary and $L = \mathrm{Imm}(\bigsqcup_{i=1}^n S^1,\Sigma)$ the space of all generic (i.e. transversally and at most doubly intersecting) immersions ...
Qwert Otto's user avatar
9 votes
2 answers
547 views

Is a local system on a surface determined by simple closed loops?

Let $\Sigma$ be a closed oriented topological surface of genus $g\geq 2$, and let $\mathfrak{X}_n$ denote the $\mathrm{SL}_n$-character variety of $\pi_1(\Sigma)$, i.e. $$ \mathfrak{X}_n= \mathrm{Hom}(...
Josh Lam's user avatar
  • 222
3 votes
0 answers
92 views

Relation of geometric and polyhedral convergence

By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
bergfalk's user avatar
2 votes
1 answer
323 views

Correspondence between fundamental group and geometric properties of $X$

At the time of studing some algebraic topology I was wondering about the following. Let $X$ be a topological space and $\pi_1(X)$ be its fundamental group. If we assume some algebraic property of $\...
KAK's user avatar
  • 321

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