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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Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible.

Show that Riemannian manifold $M$ is contractible if the loop space $\Omega_{p,p}$ is contractible. The problem is from the following material. It contends that the result is by standard Morse ...
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0answers
33 views

Homology, addition of homology classes in construction of Poicare Sphere

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for ...
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0answers
38 views

Parallelization of a Sphere gives Division Algebra

Is there an elementary proof of the fact, that a parallelization of $S^n$ can turn $\mathbb{R}^{n+1}$ into a division algebra? My guess was something like this: Let $v_1(x),\dots, v_{n}(x)$ denote ...
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26 views

the definition of relative Thom spaces

I find the definition of Thom space on the book Characteristic classes, J.W. Milnor, J.D. Stasheff, page 205: For a vector bundle $\xi$ with a Riemannian metric over a $CW$-complex $B$, let $A$ ...
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21 views

Euler characteristic and Phase rule? Is there a connection between them?

Eulers characteristic states $$Vertices+Faces=Edges+2$$ Gibbs' phase rule states $$ (\text{degrees of Freedom}) + (\text{no. of Phases}) = (\text{no. of Components}) + 2$$
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2answers
57 views

Non-compact 3-manifold with incompressible boundary

Is there an orientable, irreducible, non-compact, 3-manifold $M$ with $\partial M\cong \Sigma_2$ , genus 2 orientable surface, with $\pi_1(M)\cong \pi_1(\Sigma_2)$ and $M$ not $\Sigma_2\times ...
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1answer
15 views

Construction of an $n$-Sphere

I have been thinking about various ways to construct an $n$-sphere. Starting with $S^2$, we can construct it by taking two disks, lifting the "meat" of the disks into a third dimension and then ...
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1answer
77 views

compactification of $\Bbb R^n$

I want to show that $n$ dimensional real projection space is compactification of $\Bbb R^n$, how can I do that? I can show PR^n is compact, and R^n is localy compact, but I have problem to show R^n ...
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66 views

Graph theory: creating surfaces

If we draw a circle in the plane , we separate the plane into 2 regions , inside and outside. On the following surfaces, determine the maximal number of circles that you can draw on the surface ...
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51 views

Compute Euler characteristic $\chi(S^2 \times S^3)$

This is a question from an undegraduate Topology course Compute the Euler characteristic $\chi(S^2 \times S^3)$ For topological polyhedra $\chi(X \times Y)=\chi(X) \times \chi(Y) \implies ...
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32 views

Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
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Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times ...
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53 views

Is this octogon topologically equivalent to the Klein Bottle?

Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered. Is the space below also a Klein bottle ...
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1answer
42 views

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
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1answer
49 views

Homeomorphism between $\Bbb S$\ $\lbrace-1\rbrace$ and $\left(-1,1\right)$

I Know that $\Bbb S$\ $\lbrace-1\rbrace=\lbrace e^{i\theta}:\theta\in \left(-\pi,\pi\right)\rbrace=\lbrace e^{i\pi t}: t\in\left(-1,1\right)\rbrace$ Let $f:\Bbb S$ \ ...
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2answers
34 views

Is an isometric and bijective mapping between two metric spaces complete?

If I have the two metric spaces $(X,d_x)$ and $(Y,d_y)$ with the mapping $f : X \to Y$ that is both an isometry and bijection between X and Y. How do I show that $(Y,d_y)$ is complete iff $(X,d_x)$ ...
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1answer
22 views

Is the fundamental group of a retract a subgroup of the original space?

Let $X$ be a topological space and $A$ a retract of $X$. Is the fundamental group of $A$ a subgroup of the fundamental group of $X$?
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1answer
23 views

Euler's formula about graphs embedded in $\mathbb{R^2}$

State and prove Euler's formula about graphs embedded into $\mathbb{R^2}$ I know that if we suppose $ G $ is a finite connected graph drawn on the surface of a sphere $ S^2 $. Then the ...
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36 views

stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$ S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2. $$ Question 1: the ...
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1answer
61 views

if M is compact and N is connected, then M=N …?

Let M and N be surfaces in $R^3$ such that M is contained in N. If M is compact and N is connected, prove that M=N. ================================= I thought intuitively the compactness means ...
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1answer
86 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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1answer
40 views

Is $\overline{u^{-1}((-\infty,a))}$ a $C^k$-manifold with boundary for a $C^k$ function $u$?

Let $u\in C^k(\mathbf{R}^d)$, $M:=\overline{u^{-1}((-\infty,a))}=\overline{\{x\in\mathbf{R}^d \mid u(x)<a\}}$, where $a$ is a constant such that $M\neq\emptyset$. Is $M$ a $C^k$-manifold with ...
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1answer
31 views

figure-8 knot complement

The figure-8 knot seen as a 2-bridge knot with two maxima and two minima of the height function, has a complement in $S^3$ with one 0-handle,two 1-handles, two 2-handles and a 3-handle which cancels ...
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1answer
24 views

Path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$

I am trying to find a path homotopy between $\alpha(t) = t-t^2$ and $\beta(t) = t^2-t$ where $t\in[0,1]$ $\alpha$ and $\beta$ are path homo topic if they have the same endpoints, $p, q$ and $\exists ...
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group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
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1answer
59 views

How to determine all graph automorphisms for a given graph?

In Farb and Margalit's A Primer on Mapping Class Groups, the proof of Theorem 3.10 uses the notion of graph automorphisms. I don't know much about graph theory and have trouble following their ...
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0answers
26 views

Can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$? [duplicate]

As the question title suggests, can $S^k \times S^n$ be embedded into $\textbf{R}^{k + n + 1}$, i.e. in codimension $1$?
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1answer
53 views

Definition of n-cell and cell complex

I am taking a class on Elementary Geometric Topology and studying Kinsey's book: Topology of Surfaces. When defining an n-cell, the words "interior","boundary" and "frontier" is used without ...
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2answers
59 views

End of 3-manifold

Let $M$ be an irreducible, orientable, open 3-manifold with finitely generated fundamental group, this gives us a Scott's Core $C\hookrightarrow M$ so that the inclusion is an homotopy equivalence and ...
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0answers
23 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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0answers
32 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 ...
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0answers
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
28 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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0answers
11 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 ...
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2answers
76 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 ...
9
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1answer
74 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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0answers
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
23 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 $$ ...
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1answer
24 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 ...
5
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1answer
63 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
28 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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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
36 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
36 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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0answers
59 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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41 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,
5
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1answer
109 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 ...
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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
76 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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38 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 ...