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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1answer
211 views

Topological Spin manifolds in dimension 4

In his ICM Adress at Nice (Proceedings of the International Congress of Mathematicians Nice, September, 1970, Gauthier-Villars, editeur, Paris 6 e ,1971, Volume 2, pp. 133-163.), Robion Kirby ...
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0answers
89 views

immersions of low dimensional projective spaces

It was proved by Hirsch in (I think) 1959 that 3-dimensional real projective space can be immersed in 4 dimensional Euclidean space. Does anyone know an explicit immersion? Ditto for immersions of 4-...
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1answer
178 views

Homology modules and symmetry

Let $B$ be a cellular (simplicial, semi-simplicial etc) complex having $\mathbb{Z}^n$-symmetry (that is, there is a free action of $\mathbb{Z}^n$ on $B$, commuting with the boundary operators) and let ...
5
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1answer
177 views

Abelianization of mapping class groups $\Gamma_{g,n}$

Let $S_{g,n}$ be a Riemann surface of genus $g$, with $n$ points removed. The mapping class group of $S_{g,n}$ is denoted by $\Gamma_{g,n}$. Is there a reference where the abelianization of $\Gamma_{...
10
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1answer
380 views

Infinite cyclic subgroups of $\text{SL}_2(\mathbb{Z})$

Let $\Gamma = \operatorname{SL}_2(\mathbb{Z})$ be the usual modular group. It is well-known that $\Gamma$ contains infinitely many distinct (non-conjugate even) subgroups which are isomorphic to the ...
4
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1answer
257 views

outer automorphism classification

I am trying to understand Bestvina's "A Bers-like proof of the existence of train tracks for free group automorphisms". I'm going to ask a probably trivial question ... Here we go: The automorphism $\...
1
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1answer
104 views

Twisted torsions of reducible representations of knot groups

Let $L$ be a link in $S^3$ and $\rho : \pi_1(S^3 \setminus L) \to \operatorname{SL}_2(\mathbb C)$ be a representation of its knot group. If the twisted homology $H^\rho(S^3 \setminus L)$ is acyclic, ...
7
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1answer
472 views

Can two small exotic smooth $\mathbb{R}^4$ manifolds be combined as a standard smooth $\mathbb{R}^4$?

I just seen De Michelis and Freedman's paper Uncountably many exotic $\mathbf{R}^4$'s in standard 4-space, J. Differential Geometry 35 (1992) pp 219-254, doi:10.4310/jdg/1214447810. If I understand ...
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1answer
106 views

Link invariants distinguishing components

I was recently thinking about links where each component plays the same role: for every permutation of components, there is an isotopy permuting these components in the prescribed way. In the vein of ...
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1answer
141 views

Non-positive sectional curvature in 3-dimensional manifold

The answer of the following question may be well-known in the field of Geometric Topology, so I ask for help in here. Does the total space of circle bundle over a closed hyperbolic surface admit a ...
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0answers
130 views

When can a spatial Manifold be deformed into another spatial manifold?

I'm reading a paper by Gaiotto, Kapustin, Seiberg, and Willett, titled "Generalised Global Symmetries" (MSN). In section 3.1, paragraph 4 they write something to the order of: If we have a $M^d$ ...
4
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1answer
185 views

One-dimensional compacta as projective limits

Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1. Is it possible to express $X$ as a filtering projective limit of finite graphs? Here finite graphs ...
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89 views

Parametric Seifert surfaces for parametric families of knots in $\mathbb{R}^3$

Let $K_t$ be certain $1-$ parametric family of knots in $\mathbb{R}^3$. I am wandering what are the precise obstructions for a parametric Seifert surface to exist; i.e. a $1-$parametric family of ...
16
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1answer
439 views

Extending a map from $S^n\to M^n$ to a nice map from $B^{n+1}\to M^n$

Let $S^n$ and $B^{n+1}$ be the unit sphere and unit ball in $\mathbb{R}^{n+1}$, and let $M^n$ be a contractible space of dimension $n$. If necessary, assume that $M^n$ is a contractible simplicial $n$-...
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289 views

How does Outer Space look like without a simplex?

Considering the simplicial structure of Culler and Vogtmanns Outer Space $CV_n$. The question is now: Let $\Delta \subset CV_n$ be a closed simplex of dimension $3n-4$ or $3n-5$, how does $CV_n \...
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92 views

Does Heegaard splitting relate topological properties of a $3$-manifold to properties of subgroups of $MCG$

In the proof of Lickorish-Wallace theorem, we use Heegaard splitting of a closed, orientable and connected $3$-manifold and obtain a surface diffeomorphism which glues the two handle-body. I wonder ...
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38 views

Smooth subdivision which is not rectilinear

What is the simplest example of a smooth subdivision of the standard simplex $\Delta^n$ which can not be realized as a rectilinear subdivision? That is I want a simplicial complex $K$ for which there ...
5
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1answer
130 views

Immersion in $\mathbb R^3$ of a Klein bottle with Morse-Bott height function without centers

Can the Klein bottle be immersed in $\mathbb R^3$ so that the associated height function be of Morse-Bott type and have no centers? That is, the height function would have only Bott-type extrema and ...
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1answer
137 views

How to define “interior” for the unit arc? [closed]

Let the unit arc be, $$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$ There is something I found curious about the unit arc which is that, It has an empty interior viewed as a ...
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94 views

Slice knot and rational ball

I'm trying to understand the proof of the following classical theorem of Casson and Gordon. Theorem (CG86): If $K$ is a smoothly slice knot in $S^3$, then its double branched cover $\Sigma(K)$ bounds ...
3
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1answer
192 views

Immersion of non-orientable surface in $\mathbb R^3$ with conditions on the height function

EDIT: The answer is trivially positive; the question arose from my misunderstanding of the figure below. Can a non-orientable closed surface of odd genus be immersed in $\mathbb R^3$ so that the ...
3
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1answer
82 views

Does every locally compact connected homogeneous metric space admit a vertex-transitive 'grid'?

This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this ...
3
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3answers
168 views

Continuous map on the simplex (applicable to fair division)

Let $g$ be a continuous function from the unit simplex $D(n)$ (with $n$ vertices) into itself, that leaves invariant its vertices, and such that $g$ is not onto: to fix ideas say that $g(D(n))$ does ...
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376 views

The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?

Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
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1answer
141 views

Algorithm for identifying reducible braids

If $\vec{n} = (n_1,...,n_k)$ is a vector of integers, there seems to be a well-defined homomorphism $B_k \ltimes \left(B_{n_1} \times \cdots \times B_{n_k}\right) \to B_N$ where $N = \sum n_i$ and $...
5
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0answers
145 views

Two papers on surface diffeomorphisms

The following two papers appeared in the reference of a paper i was reading.It seems that neither is published formally.Is there a website where i could find them? A. Casson, Cobordism Invariants of ...
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250 views

When do surfaces in $\mathbb{R}^4$ intersect all their translations in one direction?

I am looking for research or references on the following problem. Let $S$ be a smoothly embedded connected surface in $\mathbb{R}^4$, with or without boundary. Fix some axis in $\mathbb{R}^4$, let $d ...
10
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2answers
508 views

Road map to learn about $\mathrm{Out}{F_n}$

I'm a last year undergraduate student and I have taken a graduate course in geometric group theory. I'd like to start reading some more advanced stuff in geometric group theory and in particular ...
7
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1answer
111 views

Extending a triangulation of the boundary of $M \times I$

(Sorry for what is probably a rather foundational PL-topology question.) By a triangulation of a manifold $M$, I mean a homeomorphism with the geometric realization of a simplicial complex, $h: |K| \...
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0answers
90 views

Khovanov Homology in Macaulay2

Has anyone ever written code for computing Khovanov homology in Macaulay2 or other similar software? I know there are various excellent programs for computing Khovanov homology, but I'm currently ...
2
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1answer
276 views

Euler Characteristic of $SL_m(\mathbb{C})/SO_m(\mathbb{C})$

As described in the title, what is the (topological) Euler characteristic of the homogeneous space $SL_m(\mathbb{C})/SO_m(\mathbb{C})$?
7
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2answers
147 views

Existence of n-axial elements in groups with at least 2 ends

Let $G$ be a finitely generated group. Fix some symmetric finite generating set $S$ for $G$, and write $\Gamma$ for the Cayley graph of $G$ with respect to $S$. Given finite subsets $X,S,Y$ of $G$, ...
4
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0answers
110 views

Are these two arguments incompatible?

I want to understand why (if so) these two arguments are not incompatible. And if that's the case, which one is wrong. First we have this paper (by Honda, Kazez and Matic). We look at the last Lemma ...
5
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0answers
133 views

Description of quasimorphisms of the free group

Let $F$ be a free group of finite rank with a fixed basis and corresponding word metric. Let $Q = Q^0_h(F, \mathbb{R})$ be the space of real homogenous quasimorphisms that vanish on the basis of $F$. ...
5
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2answers
156 views

Reference request: Reidemeister type moves for immersed curves on surfaces

Preliminaries Let $\Gamma$ be a closed $1$-manifold (i.e. a union of finitely many circles) and let $\Sigma$ be a closed $2$-manifold (i.e. a surface). I'll adopt the following terminology. ...
5
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1answer
230 views

Relations between boundaries of groups acting on hyperbolic spaces with WPD elements

Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions. Say that the action is acylindrical if ...
3
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1answer
268 views

Lectures on triangulations of manifolds by Robion Kirby

I was looking for the book mentioned in the title. Seemingly it was not published, but copies are available in several mathematical libraries. Google books does not provide preview. I am wondering if ...
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0answers
91 views

Understanding $w_2$ as an obstruction to trivializing the tangent bundle over 2-cells

I am reading through "A Geometric Proof of Rochlin's Theorem", and it is occurring to me, again, that I don't understand spin structures / $w_2$. My confusion arrises in, naturally, the proof of ...
4
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0answers
116 views

Borromean Lines of three $\mathbb{R}^1$ in $\mathbb{R}^3$ and analogous Milnor link invariants

It is know that Borromean rings can be detected by Milnor invariant $$ \bar{\mu}(\gamma_1,\gamma_2,\gamma_3)= \# (\Sigma_1 \cap \Sigma_2 \cap \Sigma_3)-\frac{1}{2}\sum_{I,J,K}\epsilon_{IJK} \sum_{\...
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0answers
100 views

When do unknots in $S^n$ bound unique balls?

I recently heard the it is an open question whether or not an unknotted $S^2$ in $S^4$ bounds a unique $B^3$ in $S^4$, where by unique, I mean up to isotopy rel boundary in $S^4$. The rel boundary ...
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0answers
123 views

A topological property of curves on the plane $\mathbb{R}^2$

Let $\gamma\colon [0,1]\to \mathbb{R}^2$ be a continuous injective map. Is it true that for any inner point $t\in (0,1)$ there exist an open neighborhood $U$ of $\gamma(t)$ and a homeomorphism $f\...
4
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0answers
62 views

Proximal isometries in CAT($-1$) metric space

Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...
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0answers
15 views

Is the Set of all pairwise continuous maps from a bitopological space to a finite heyting algebra,$(L,T,T)$ forms heyting algebra?

As we know the Set of all continuous functions from a topological space to a discrete space(L, T) forms a Heyting algebra, where L is finite distributive latices i.e., a Heyting algebra. So the ...
4
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0answers
227 views

Is there a one-relator circle-packing theorem?

Let $X_w$ be the presentation complex of a one-relator group $\langle x_1,\dotsc,x_n\mid w\rangle$, with $w$ cyclically reduced, i.e., $X_w=R\cup_w D$, with $R$ the rose with $n$ petals labeled $x_1,\...
10
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1answer
355 views

Piecewise linear Poincaré conjecture

Let $M$ be a PL-manifold that is a homotopy sphere (PL stands for Piecewise Linear). Does it follow that $M$ is PL-homeomorphic to the sphere $S^n$ (with the usual PL-structure)? Here is the ...
4
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2answers
225 views

Maps between grassmannians with inclusion property

Edit: According to the comment of L. Spice I changed the inclusion sign to the subset sign. Is there a continuous map $f:\mathbb{C}P^3 \to \textrm{Gr}_{\mathbb{C}}(2,4)$ with $x\subset f(x)$? What ...
2
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0answers
72 views

Cohomology of colored braid groupoids

Consider braids on $n$ strands and pick $n$ distinct labels $1, \dots, n$. There is a groupoid $\mathcal P_n$ whose objects are tuples $(l_1, \dots, l_n)$ of labels and whose morphisms are braids, ...
4
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0answers
134 views

Symmetries of MCG in terms of Humphries generators?

The Riemann-Hurwitz formula gives $84(g-1)$ as the upper bound to the order of a finite group acting faithfully on a closed genus g surface. Famously the bound is realized when $g=3$ by a simple ...
4
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1answer
236 views

On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1]. In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
7
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1answer
228 views

Homotopy in $X$ and homology in $X \times I$

Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f_1,f_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f_1$ and $...

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