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
143 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 ...
2
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2answers
144 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 ...
2
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1answer
47 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
166 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 ...
17
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1answer
357 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 ...
2
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0answers
23 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
52 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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0answers
175 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
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1answer
57 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
153 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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1answer
216 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
96 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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1answer
68 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 ...
4
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0answers
62 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
328 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 ...
8
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1answer
119 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 ...
3
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2answers
199 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
98 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
71 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 ...
4
votes
1answer
154 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 ...
48
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0answers
1k 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. ...
6
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1answer
324 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 ...
6
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1answer
107 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
votes
1answer
82 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
112 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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votes
0answers
77 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 ...
5
votes
1answer
141 views

Showing every knot has a regular projection using differential topology

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 imbedding. For ...
0
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2answers
77 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 ...
6
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2answers
380 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
votes
1answer
123 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
votes
1answer
132 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 ...
3
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0answers
114 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
47 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
134 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 ...
17
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2answers
821 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
431 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
votes
3answers
164 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
90 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 ...
10
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1answer
370 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
452 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 ...
2
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1answer
86 views

Uniqueness of “Punctured” Tubular Neighborhoods (?)

Here is a question that has been haunting me for a while: Let $\mathbb{R}^{n-1} \times [0, \infty)$ be the upper half space of $\mathbb{R}^n$ and suppose we have a smooth homeomorphism (not a diffeo) ...
3
votes
1answer
316 views

Annulus Theorem

I'm trying to read Rolfsen's "Knots and Links" and I'm a little discouraged that I can't do one of the first and seemingly more important exercises. The question is Use the Schoenflies theorem ...
2
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0answers
95 views

$\epsilon$-net of a $n$-dimensional $\ell_2$-ball

Let $B$ be an $\ell_2$-ball of radius $r$ in $\mathbb{R}^n$. I want to find the cardinal of a (not too big) $\epsilon$-net of $B$, that is the cardinal of a finite set $V\subset B$ such that $\forall ...
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0answers
63 views

Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov

Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
1
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1answer
229 views

Is there an example of a non-orientable group manifold?

Basically what I'm looking for is a topological group that is also a non-orientable, n-dimensional manifold
4
votes
1answer
417 views

3-manifolds fibering over the circle and mapping tori

If $S$ is a closed connected surface and $\varphi \in \mathrm{Diff}(M)$, then we can build the mapping torus $M_\varphi = \dfrac{S \times [0,1]}{(x,0)\sim (\varphi(x),1)}$. Then we have that $ ...
2
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0answers
72 views

PL and triangulizable

Is it correct that the notion of triangulizable manifold (in the sence "homeomorphic to a simplicial complex") is weaker than the notion of a PL-manifold? If yes, why? (eg is it true that a star of ...
3
votes
1answer
345 views

What is the proof that SO(2n+1) is non-orientable for any positive integer n?

Inquiring minds want to know. :) I know for certain that SO(3) is not orientable but I did read somewhere that for any odd dimension N>1, SO(N) is a non-orientable manifold. If such is true I'm eager ...
2
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0answers
79 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
1
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1answer
284 views

Immersing punctured torus

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...