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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25 views

Orientation of a triangulated compact surface, using orientations of triangles

The questions I am working on asks me to :"Give the definition of an orientation of a triangulated compact surface by using orientations of triangles" I know that a surface is orientable if the ...
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40 views

proper submersion

I have the following question: Let $X,W$ be smooth manifolds with $W\subset X\times \mathbb{R}\times \mathbb{R}^n$ and the projection $p_{1}:W\rightarrow X$ a surjective submersion. Let $a:X\...
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19 views

Shrinking some polygons to make the containing polygon connected

Inside a public area $C$ (a polygon), there are several private land-plots $C_1,\dots,C_n$ (pairwise-disjoint simple polygons): Currentlly, the public area that is outside the private land-plots (...
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1answer
29 views

Evaluating Colored Jones Polynomial of a trefoil knot

Following arXiv:1211.6075v1 I want to calculate colored Jones Polynomial for trefoil knot. I have the formulas: $ J_{\oplus R_i} = \sum_i J_{R_i} (K,q)$ $J_{R} (K^n, q) = J_{R^{\otimes n}} (K,q)$ I ...
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12 views

Reference request for skein modules

I'd like a basic (as basic as possible :)) reference to see basic definitions and properties of Skein modules and the Kauffman relations for three manifolds. (I'm trying to understand a survey (https:/...
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2answers
78 views

Surface filling simple closed curves

At least how many simple closed curves do we need to fill a surface? Definition: Let $A$ be set of simple closed curves. $A$ is filling set if all other curves on the surface that are not parallel ...
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1answer
78 views

In how many dimensions is the full-twisted “Mobius” band isotopic to the cylinder?

There is a question on this site about the distinctions between the full-twisted Mobius band and the cylinder, but I would like to ask something different, so I start a new question. Let us call $C$ ...
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50 views

About $T^6$ and $T^2 \times T^2 \times T^2$

I read often that wen can see the six torus like $T^2 \times T^2 \times T^2$. So, what is the difference between $T^6$ and $T^2 \times T^2 \times T^2$ ?
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1answer
31 views

Bockstein homomorphism and the universal coefficient theorem

The following statement is given in the third comment of kernel of the mod $2$ Bockstein on the first cohomology group: Statement: Let $X$ be a path-connected finite $CW$-complex. Suppose $$ H_1(X;...
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1answer
27 views

Open mapping and space filling

Suppose $n<m$. Is there a continuous function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ that is also an open mapping (maps open sets to open sets)? For example, are any of the standard space-filling ...
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1answer
64 views

Minimal-dimension example of (open) subset of $\mathbb{R}^n$ with trivial first cohomology but nontrivial fundamental group

As a follow-up to this question, I was wondering what dimension provides the minimal counterexample to the claims: If $U\subseteq\mathbb{R}^n$ is an open connected set with trivial $H^1(U)$, then $\...
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2answers
30 views

Number of distinct dense subsets

Let's consider $X$ a topological space. Let's say that $X$ is hausdorff but not compact. Can there be in $X$ two disjoint dense subsets $S_1$ and $S_2$? If the answer is yes, then is there a limit to ...
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0answers
40 views

Part2: Two ambient isotopic curve segments, one has the length and the other does not

Let me start with an R^2 ambient isotopy J taking a straight line C1 to some C2. An answer of other question implies that it can happen that you cannot define the length for C2. [Answer](Two ...
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1answer
39 views

Two ambient isotopic curve segments, one has the length and the other does not

Let me ask if the following is possible: Let $L_1$ be some curve segment in the $\mathbb{R}^3$ space which has the length $1$. Let $L_2$ be some curve segment in the $\mathbb{R}^3$ space which you ...
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2answers
39 views

Euler characteristic of a sphere using 'grid method'

If I place the following grid (see picture) on the sphere, how can this determine its euler characteristic? I know that the formula for the Euler characteristic for surfaces is: $E=V-E+F$ with the ...
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62 views

Computing fundamental group. Using Van Kampen? Visualization of a space

I'm trying to compute the fundamental group (using Van Kampen) of a space which appears when identifying the disjoint boundaries of a 3-manifold with boundaries. My knowledge of 3-manifolds is none ...
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43 views

Triangulation of $S^2 \times S^2 $

Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ? Thanks,
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1answer
117 views

Characterization of open sets in $R^3$ homeomorphic to $R^3$.

Background: By the Riemann mapping theorem, for any non-empty, simply connected open subset $U \subset \mathbb{C}$, $U \neq \mathbb{C}$ there exists a biholomorphic map (in particular a homeomorphism)...
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21 views

A disk cross sphere in a sphere

Let $\lambda$ and $\mu$ be positive integers and define $D(\lambda, \mu) = \{ p \in S^{\lambda + \mu -1} : \sum_{i \leq \lambda} x_i^2 \geq \sum_{i > \lambda} x_i^2 \}$. Why is $D(\lambda, \mu)$ a ...
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2answers
81 views

Prerequisites for Kirby Calculus?

I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus? I have a solid ...
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0answers
47 views

Gluing two solid tori by a homeomorphism of their boundarries.

I am aware that we get all lens spaces by gluing two solid tori by their boundaries. My question is, do we get more spaces besides lens spaces? in other words, do all homeomorphisms of the boundaries ...
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0answers
27 views

The condition to ba an ambient isotopy

Let me ask the following question about ambient isotopies. https://en.wikipedia.org/wiki/Ambient_isotopy An article in Wikipedia says: A continuous map F:M * [0,1] -> M is defined to ...
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21 views

Diffeomorphisms with fixed subsets and isotopies

I asked a much more general version of this question previously, but I realize now that it was so broad that there probably are no good general answers, so I want to make a more specific question. ...
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1answer
26 views

Isotopies with fixed subsets

Let $M$ be a smooth manifold. Let $f:M\rightarrow M$ be a diffeomorphism which is smoothly isotopic to the identity. Let $X\subset M$ be a compact subset such that $f|_X = id_X$. Under what ...
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1answer
55 views

Reference on manifolds with corners

Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) ...
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1answer
51 views

Estimate on Size of Convex Hull

Let $X$ be a bounded subset of $\mathbb R^3$, and denote by $Conv(X)$ the convex hull of $X$, i.e, the smallest convex subset of $\mathbb R^3$ containing $X$. As $X \subset Conv(X)$ by definiton, the ...
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1answer
44 views

Smooth vector field vanishing at exactly 6 points

Let $ E $ be the ellipsoid in $ \mathbb{R}^3 $ given by, $$ E = \left \{ (x,y,z) \mid \frac{x^2}{9} + \frac{y^2}{4} + z^2 = 1 \right \} $$ Find a smooth vector field $ H : \mathbb{R}^3 \rightarrow \...
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1answer
58 views

Extension of Smooth Functions on Embedded Submanifolds

In Lee Smooth Manifolds, this problem is given: if $S \subset M$ is smoothly embedded and every $f \in \mathcal{C}^{\infty}(S)$ extends to a smooth functional on $\textit{all}$ of $M$, then $S$ is ...
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0answers
36 views

Fundamental group from triangulation

How do I calculate the fundamental group of surface if I have been given it in triangulated form. I have attached an example triangulation I think I may have to simplify it further to a tree and ...
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1answer
30 views

Show that chordal metric is topologically equivalent to the Euclidean metric

Consider $$d(x,y)=\frac{2\|x-y\|}{(1+\|x\|^2)^{1/2}(1+\|y\|^2)^{1/2}},\hspace{5mm}x,y\in \mathbb{R}^n.$$ $d$ is a metric in $\mathbb{R}^n$ known as chordal metric. I want to show that this metric is ...
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546 views

Why are invariants of knots and manifolds important or useful?

It is easy to define invariants that completely classify knots; however, this is computationally infeasible, so is it computationally efficient invariants that are important? Why? Do mathematicians ...
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39 views

Relation between Alexander duality and linking numbers

I just have the feeling that there must be some relation between Alexander duality and linking numbers, but I don't know what is that. Will anyone tell me anything about that? Or could anyone give ...
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0answers
37 views

Does this container exist?

EDIT: Note that the object I'm seeking needn't have anything to do with water or actual containers; those are just used to convey the idea. I'm trying to find a container that, when turned with ...
0
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1answer
48 views

Knotted cross-sections of unknotted spheres

Livingston's notes on concordance mention "embeddings of $S^2$ into $\mathbb{R}^4$ which are unknotted, but have non-trivial knots as cross-sections. There are other such unknotted two spheres with ...
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1answer
136 views

Proof of the bigon criterion

In Farb and Margalit's A Primer on Mapping Class Groups we have the following Proposition 1.7: Two transverse simple closed curves in a surface $S$ are in minimal position iff they do not form a ...
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36 views

universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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1answer
79 views

Connected sum of two “same” Klein bottles

If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles? This is what ...
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3answers
43 views

What is the distance of two circles put on eachother?

If one takes two circles (lets say on a straight cilinder), and bring the circles closer and closer to eachother. Will the distance between them goes to zero, or can you say the distance is for ...
1
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1answer
44 views

Unbounded, closed, star-shaped set contains ray

I am trying to prove the following statement: Let $R$ be a real closed field (such as the real numbers). Let $M\subseteq R^n$ be a semi-algebraic set, i.e. a set which is defined by a Boolean ...
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0answers
76 views

How to prove, no tame knot is isotopic to a non tame knot?

Please let me ask the following question. I have read in Wikipedia, quote: A polygonal knot is a knot whose image in R^3 is the union of a finite set of line segments. A tame knot is any knot ...
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0answers
32 views

A. A. Markov's paper on insolubility of the homeorphy problem [duplicate]

I am aware that this has been asked before, but the paper is nowhere to be found online, the provided link in the old thread leads to nowhere, and I'm really at wits end to find this paper, can anyone ...
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1answer
100 views

Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...
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0answers
60 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
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78 views

Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ (x,0)...
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1answer
60 views

Regular neighbourhoods of non-orintable surfaces in $S^4$

Suppose that $F \subset S^4$ is a non-orientable surface. Let $N \subset S^4$ be a regular neighbourhood of $F$. Clearly, the boundary of $N$ is a circle bundle over $F$. Which is its Euler number? My ...
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42 views

Simply connected compact subsets of $\mathbb R^2$

Is every simply connected compact subset of $\mathbb R^2$ weakly contractible, i.e. all homotopy groups vanish?
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39 views

Plane curves admitting several ways to seal it up by the disc

Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ ...
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1answer
56 views

Components of the space of immersions 2-manifold into $\mathbb R^3$

Let $M$ be a $2$-sphere with $g$ handles. Consider the space of maps $M\to \mathbb R^3$, which are immersions [i.e. smooth maps with nondegenerate differential in each point $x\in M$], with compact-...
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0answers
50 views

What is the fundamental group of a cone?

I am reading an article on orbifolds and it describes the cone as the quotient of unit 2-dim. disc by a finite cyclic group of rotations. But how is it's fundamental group finite cyclic?
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40 views

Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is ...