The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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1answer
50 views

Definition of “Representing” a Handlebody (Lefschetz Fibration)?

Sorry, I could not find a clear explanation of the meaning of the word represented in the following:"any 4-dimensional 2-handlebody W can be represented by a topological (achiral) Lefschetz fibration ...
3
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2answers
44 views

Identification Space and Isotopy

Original Question: Let $X$ and $Y$ be topological spaces and let $f:X \to Y$ and $g:X \to Y$ be isotopic embeddings. Is it true that $X \cup_f Y$ is homeomorphic to $X \cup_g Y$? Edit: I meant to ...
4
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1answer
63 views

What's the difference between “crumpled cube” and “3-ball”?

Warning: My level of understanding of topology is very low. Small words would be appreciated. :) Browsing Wikipedia, I came to crumpled cube, defined as "a 2-sphere together with its interior". ...
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1answer
418 views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a ...
2
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1answer
61 views

Homologous surfaces in three-manifolds

Let M be a 3-manifold. Let $S$ and $T$ be properly embedded surfaces in $M$ such that $[S] = [T] \in H_2(M, N(\partial S)) $. Is it true that we can isotope $\partial S$ so that it coincides with ...
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1answer
60 views

Torsion Subgroup of Mapping class group.

What is the cardinality of finite order elements in Mapping class group of a surface $S_{g,n}$ of genus g and n boundary components. 1) If it is infinite then how can I generate a collection of ...
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2answers
88 views

Quasi-isometric embedding and Quasi-isometry

Let $X$ and $Y$ be geodesic metric spaces. Suppose there are quasi-isometric embeddings $f:X \rightarrow Y$, $g:Y \rightarrow X$. Then, can we say there is a quasi-isometry from $X$ to $Y$? I tried to ...
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0answers
27 views

Calculation of First Fundamental Form for an aribtrary Surface given as a graph

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and let $X \in C^2( \bar{\Omega} \, ; \mathbb{R}^{n+1})$ and $X(\theta) := (\theta, h(\theta))$, $\theta \in \bar{\Omega}$ be a parametrization ...
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1answer
36 views

Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let (Y,ξ) be a contact manifold, and L⊂(Y,ξ) be a Legendrian knot which is the boundary of a convex surface Σ embedded properly in Y. Why ...
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2answers
97 views

Frame bundle of orthonormal frames orthogonal to a submanifold.

Suppose we have a smooth manifold $M$ of dimension $m$ with a Riemannian metric and a connected submanifold $N$ of dimension $n$ in $M$ with $n<m-1$. Let $n\le k<m-1$ and consider the bundle ...
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0answers
27 views

Intersection of closures of the Schubert cells

How can I determine does two closures of Schubert cells of Grassmannian $Gr(n,m) $ $e(\sigma_1,...\sigma_m)$ and $e(\sigma_1',...\sigma_m')$ intersect or not.
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1answer
295 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
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0answers
21 views

Edges and genus in graphs

For a planar graph $G = (V, E)$ there is the well known bound $|E| \leq 3|V| - 6$. If instead of $S^2$ $G$ embeds in the orientable surface $S_g$ of genus $2 - 2g$ with minimal $g$, what can be said ...
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1answer
41 views

The deficiency of surface group

Let $G$ be the fundamental group of a closed surface of genus $g$. We know $G$ has a presentation $$\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\dots[a_g,b_g]=1 \rangle.$$ The ...
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1answer
50 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
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1answer
104 views

The image of homomorphism of fundamental group of closed surface

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus $\geq 2$. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to ...
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2answers
357 views

Simply connected does not imply contractible. Is there a nice counter example in $R^2$?

The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were ...
5
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1answer
101 views

How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
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0answers
25 views

Theorems in Topology [duplicate]

Is every theorem that is provable in one form of topology also provable in another form of topology, say all theorems in algebraic topology are also provable in point set topology or computational ...
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73 views

3-Manifolds from identifying faces

I have a question about two polyhedra and getting manifolds out of them. The first of these is a tetrahedron. When calculating the Euler characteristic I got $\chi(X)=1-3+2-1=-1$. I believe 3 edges ...
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0answers
26 views

Understanding the relationship between the genus and closed curves on a surface.

Recently I was told that the genus of an orientable surface is related to the number of closed curves one can remove before its disconnected. To sort of prove it to myself I drew out a 3-torus. I ...
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1answer
45 views

Boundary of a compact 3-dimensional manifold with boundary is a compact manifold of 2 dimensions.

I have been able to prove to myself that the boundary of a 3-dimensional manifold is indeed a compact set. I am stuck however proving that it is a 2 dimensional manifold. Specifically why the ...
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0answers
34 views

Extrapolating a formula from the Euler characteristic

Let $X$ be a compact surface with $\tau$ as its triangulation with $k$ vertices. Define degree to be the number of vertices originating at a vertex v. Then $$6\chi(X)=\sum\limits_{n=1}^k ...
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0answers
49 views

What is a 2-surgery on a disk?

I am confused by a certain point in Scharlemann's paper "Sutured Manifolds and Generalized Thurston Norms", which seems important enough to not just skip it. I mean the "2-surgery on disks" in the ...
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1answer
129 views

Handlebody decomposition and intuition

I am trying to get an intuition of how to approach handle body decompositions. I understand that a Torus can be decomposed into a 1 0-handle, 2 1-handles, and a 2-handle. The 0-handle is a hole you ...
7
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1answer
110 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
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2answers
140 views

Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
2
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0answers
67 views

Sufficient conditions for quasi-isometric embeddings of Cayley graph

I would like to know more about the assumptions under which the Cayley graph of a given group embeds quasi-isometrically into the space where the group is acting. For instance, if a group $G$ acts by ...
0
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1answer
60 views

Deforming disks into other disks

Right now I'm casually reading through Carson's "Topology of Surfaces, Knots, and Manifolds." I don't have a strong background in topology, and I was told that this was a very accessible and ...
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0answers
91 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
14
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270 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
4
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1answer
153 views

How can I tell if two functions are conjugates in the homeomorphism group of $\mathbb{S}^n$?

Suppose we have two functions $f,g:\mathbb{S}^n\to\mathbb{S}^n$ which are bijective, continuous, and have a continuous inverse (aka bicontinuous). They are conjugates in the homeomorphism group when ...
4
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1answer
79 views

How hard is it to endow a $\textit{Spin}^{c}$ structure on four-dimensional manifolds?

I am in a certain math conference and we came across Seiberg-Witten equations. Since I am really novice in the field, I asked if all "reasonable" four manifolds carry a $\textit{spin}^{c}$ structure. ...
3
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1answer
72 views

Cohomology calculation for maps to the 2-sphere.

Let $Y^3$ be a closed 3-manifold and $f\colon Y\to \operatorname{SO}(3)$, $g\colon Y\to S^2$ be smooth maps. Define $g'\colon Y\to S^2$ be the following composition: ...
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1answer
62 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, ...
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0answers
55 views

Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
4
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1answer
81 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
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2answers
68 views

Surgery on $S^m$

On page 4 of the book "ALGEBRAIC AND GEOMETRIC SURGERY" by Andrew Ranicki, after the definition of surgery has written: Example View the $m$-sphere $S^m$ as $$S^m=\partial (D^{n+1} \times ...
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2answers
326 views

Uniqueness of Preferred Framing of a Solid Torus in $S^3$

One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...
4
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1answer
94 views

$6n\pm 1$th fold cyclic covers of $S^3$ branched over the trefoil.

This questions is actually exercise 10D4 from Rolfsen's Knots and Links. In example 8D7 Rolfsen computes a presentation matrix for $\Sigma _n$ the n-fold cyclic cover of $S^3$ branched over the ...
7
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1answer
118 views

Loop space and stable homotopy theory

The Bott periodicity theorem for unitary group $U(n)$ says that $$ \pi_{i-1}(U) \simeq \pi_{i+1}(U) $$ How can I prove, using this theorem, that $$ \Omega (U) \simeq BU \times \mathbb{Z} ?$$ What is ...
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0answers
70 views

What is the “Standard” Open Book Decomposition for $\mathbb R^n$, and why does this matter?

I am trying to understand better Open book decompositions. To that effect, I tried to work out a couple of (relatively-simple) examples, specifically, for $\mathbb R^2 $ and higher. But I have not ...
0
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1answer
45 views

Prove $f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$

I need to show that $$f\colon X/{\sim} \to Y \text{ is continuous} \iff \pi\circ f\colon X \to Y \text{ is continuous}$$ where $X/{\sim}$ is a quotient topology and $\pi$ is the quotient map. I ...
1
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1answer
70 views

How can you prove that the winding number around two zeros of a vector field is the sum of the two indices?

If v is a continuous vector field with two isolated zeros, then the winding number around one zero is its index. The winding number on a circle with both zeros in its interior is the sum of the two ...
14
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2answers
433 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
4
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0answers
352 views

The Birman–Hilden Theorem and the Nielsen–Thurston classification

So this post is half question/half reference request, as I'm sure it's the kind of thing people would have thought about before (and indeed the question might even be trivial), but I've been unable to ...
5
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3answers
132 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
2
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1answer
80 views

Density of continuous knots in the plane transversal to some circles

This is an exercise from the book "Knots and Links" by Rolfsen (exercise 6 in section 2C) Let $\kappa : S^1 \rightarrow \mathbb{R}^2-(0,0)$ be a continuous imbedding. Let $M := \{ x \in ...
8
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1answer
158 views

Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
4
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0answers
222 views

Soft question: why are there non-smooth manifolds?

Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by ...