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Questions tagged [geometric-topology]

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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23 views

Proof for sphere eversion

Do you know where I can find (preferably online) a proof for the possibility of a sphere eversion?
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16 views

Any spin $M^3$, exists a natural induced $\text{Pin}^-$ structure on Poincare dual PD

It seems that "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural ...
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Exercise 1.8.6 - Differential topology (Guillemin and Pollack)(2)

The question and its answer is given below: My Questions are: 1- The solution has proved that $p \circ v^*$ is the identity on X and not $p \circ v$, does not that mean that the vector ...
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30 views

Twisting the unit square n times before gluing( 2.1.6 in G&P).

The question is given below: I have made a Mobius band with a paper and twisted it 3-times but I could not describe what I see it may be a 3 knot shape, could anyone give me a hint for solving that ...
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1answer
20 views

Understanding noncyclic covers of knot complements

Let $K$ be a knot in $S^3$. I'm familiar with the process of taking the $n$-fold cyclic cover $X_n(K)$ of the knot complement $X(K) = S^3 - K$ and I know that this cyclic cover can be completed to a ...
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1answer
74 views

A difficulty in understanding the solution of problem 1.7.17 in Guillemin and Pollack.(p.47)

The question is given below: And here is exercise (16): And here is the solution to exercise(17) But I have difficulties in understanding the following parts of the solution: 1-Why the codomain of ...
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2answers
15 views

Expansion of homeomorphism outside a disk

The following is an exercise in Bloch's Intro to Geometric Topology Let $B \subseteq \Bbb R^2$ be a set homeomorphic to the closed unit disk and $h :\partial B \to \partial B$, a homeomorphism. By ...
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1answer
45 views

Prove that $f$ is Morse function if an only if $det(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0$(2)

I was trying to solve this question: Prove that $f$ is Morse function if an only if $\mathrm{det}(H)^2 + \sum_{i=1}^k (\frac {\partial f}{\partial x_i})^2>0.$ But while searching on this site I ...
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1answer
31 views

Show that all open sets in $\mathbb{R}^n$ are the same for metrics $d_1,d_2,d_\infty$

So I have shown that any $x,y\in\mathbb{R}^n$ that $d_\infty(x,y) \le d_2(x,y)\le d_1(x,y)\le nd_\infty(x,y)$, where $$d_\infty=\max\limits_{j\in\{1,\dots,n\}}\lvert x_j-y_j\rvert,$$ $$d_2(x,y)=\sqrt{...
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1answer
29 views

Separation from family of closed and connected subsets

Let $X $ be a compact $T_0$ topological space and let $\{ A_i \}_{ i \in I }$ be a family of closed and connected subsets of $X $ and $ x\in X$ such that for each $i\in I $ there exists a closed and ...
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Confusion about Heine-Borel property on manifold.

I am studying Hopf-Rinow theorem in Petersen's book, and there is one of the four equivalent statements that confuses me. $M$ satisfies the Heine-Borel property, i.e., every closed bounded set is ...
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1answer
23 views

A difficulty in understanding a paragraph in p.41 in Guillemin & Pollack.

Here is the paragraph that I do not understand the last statement in it: Could anyone explain it for me please?
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1answer
38 views

About a proof using complex numbers that the antipodal map on $S^n$, $n$ odd, is homotopic to the identity map

This answer gives a solution to the following problem: Prove that antipodal map on $ S^n $ homotopic to identity map if $n$ is odd. I am wondering: Why we had to rely on complex numbers in ...
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Poincaré dual 3-manifold inside a product of real projective spaces

Consider a 5-manifold as a product of real projective spaces: $$ \mathbb{P(R)}^3 \times \mathbb{P(R)}^2, $$ Let us take a 2-submanifold as a 2-torus $T^2$ embedded in $\mathbb{P(R)}^3 \times \mathbb{...
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1answer
43 views

semi-direct product between manifolds

question 1: Are there mathematical definition of the semi-direct product between manifolds $$ M^{d_1} \rtimes V^{d_2}? $$ For example, is it defined as a fibration such that $M^{d_1}$ is the fiber ...
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1answer
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Q.1.6.1 in Guillemin and pollack.

The question and its answer is given below: But I do not understand why in the second line of the solution $\widetilde{F}$ is defined as this, could anyone explain this for me please?
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2answers
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Even dimensional indecomposable simply-connected closed manifolds with a free action of $\mathbb{Z}/4$

Proposition 2.29 of Hatcher's Algebraic Topology asserts that $\mathbb{Z}/2$ is the only nontrivial group that can act freely on even dimensional spheres. My question: Which even dimensional ...
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1answer
25 views

Removing a closed subset from a surface gives a surface

In Ethan Bloch's "A First Course in Geometric Topology and Differential Geometry" the definition of a surface is the following: A subset $Q \subseteq \mathbb{R}^n$ is called a surface if each point ...
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31 views

How to deduce Jordan Curve Theorem from Schönflies Theorem

Recently I started reading Ethan Bloch's "A First Course in Geometric Topology and Differential Geometry" and I came upon this exercise to deduce the Jordan Curve Theorem from the Schönflies Theorem: ...
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1answer
39 views

2-Simplex.. filled or not filled?

I've seen some authors define the 2-simplex as the boundary of a triangle and others define it including the interior of the triangle (i.e. filling in the triangle). Does this distinction matter? Are ...
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1answer
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Is the equivalence between a single point on a mobius strip and an unordered pairs of points on a loop unique?

This video shows the visualization of the proof of inscribed rectangular problem. It elaborates the equivalence between a pair of points on a loop and a single point on the mobius strip. https://www....
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Showing that the diagonal of $X\times X$ is transversal to the graph of $f$. (1.5.10 Guillemin and Pollack)

The question and its answer is given below: But I am wondering, is it also correct if I showed that graph f is transversal to diagonal of $X\times X$? Also, I can not understand the general ...
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1answer
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Determine if two nodes in a hierarchy are connected

I have a bunch of nodes arranged in a hierarchy structure as follows: I would like to determine if one node is connected to another node, even if the connection between the two is separated by ...
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1answer
45 views

Homology and embedded surfaces of genus $g$

Suppose to have a closed manifold $M$ of dimension $n\geq 3$ and suppose that there is an element in Homology $[S]\in H_2(M,\mathbb{Z})$ that is represented by a surface $S$ homeomorphic to the 2-...
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1answer
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Non-homeomorphic structures and the Descartes' theorem

Some structures like the donut are not homeomorphic to a sphere. According to this link (https://en.wikipedia.org/wiki/Angular_defect#Positive_defects_on_non-convex_figures) the basis of the Descartes'...
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1answer
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Can a closed surface of genus $\geq$ 2 be embedded in a product of graphs?

Let $S$ be a closed orientable surface of genus $g \geq 2$. Is there an embedding of $S$ into the product of two graphs $G_1$ and $G_2$? I can't think of such an embedding but I don't know any ...
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1answer
56 views

Realizing a CW Complex as an Adjunction Space: Munkres' Proof

Suppose $Y$ is a $CW$ complex, of dimension $p-1,\ \sum B_{\alpha}$ is a topological sum of closed $p-$ balls. Then, if $g:\sum \partial B_{\alpha}\to Y$ is a continuous map, the adjunction space $X=Y\...
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Exercise 1.5.1 in Guillemin and Pollack

Suppose that $A: \mathbb{R}^k \rightarrow \mathbb{R}^n$ is a linear map and $V$ is a vector subspace of $\mathbb{R}^n$. Check that $A$ is transversal to $V$ means just $A(\mathbb{R}^k) + V = \mathbb{R}...
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1answer
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Exercise 1.5.5 in Guillemin and Pollack.

The problem and its solution is given below: But I could not understand which proof the writer is taking about and how he get the third line in the proof. Could anyone explain this for me please? ...
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Exercise 1.5.7 in Guillemin and Pollack

The problem and its solution is given below: I am wondering if we can solve it without using 1.5.5 ?
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1answer
21 views

Separating disks in 3-manifolds

Let M be a (smooth or PL) connected three manifold with boundary, such that one boundary component is a sphere S. Let D be a properly embedded disk whose boundary lies on S. Must D separate M into ...
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1answer
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Transversality as an extension of the notion of regularity p.28 in Guillemin & Pollack.

The book "Differential Topology" book is explaining this in the image below: But I do not understand: In the first paragraph: The 2 statements starting from the fifth line, could anyone ...
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2answers
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How to introduce a CW structure on RP^n?

My first course in topology is going extremely fast, and does not seem like rigorous mathematics. Last lecture, we were given the definition of CW-structures, but did not do any examples. Yet we were ...
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A set is a manifold iff a subset of it is a manifold.(p.28 Guillemin & Pollack)

Does this statement in the picture tells me that "A set is a manifold iff a subset of it is a manifold.(p.28GP)" : Could anyone explain this for me please?
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1answer
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How to prove geometric progression

The given series is: $$6+3*2^{1-n}$$ Prove that this series is geometric progression. What is $a_n$ in this series? And also show if the series is convergent. I tried to: So we know that the sum of ...
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what is the meaning by words of the following transversality problem (GP 1.5.5)

The problem and its solution is given below: But I want to understand what is the problem saying in words, could anyone explain this for me?
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2answers
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Are these conditions equivalent to the definition of regular coordinate ball?

In page 15 of Lee's book "Introduction to Smooth Manifolds", there's a paragraph as follows: We say a set $B\subset M$ is a regular coordinate ball if there is a smooth coordinate ball $B'\supset \...
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1answer
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Prove that the paraboloid in $R^3$, defined by $x^2 + y^2 - z^2 =a$ is a manifold if $a>0$. why does not $x^2 + y^2 -z^2 =0$ define a manifold?

Prove that the paraboloid in $R^3$, defined by $x^2 + y^2 - z^2 =a$ is a manifold if $a>0$. why does not $x^2 + y^2 -z^2 =0$ define a manifold? Could anyone give me a hint of the solution of this ...
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1answer
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How to see the real projective plane is a Möbius band glued to a disk?

I am seeking an easily comprehended, convincing explanation that ${RP}^2$ is topologically the same as gluing the circle boundary of a disk to the edge of a Möbius band.
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Borsuk–Ulam theorem proof using Brouwer degree

I wonder if Borsuk–Ulam theorem (if $f:\mathbb{S}^n\rightarrow\mathbb{R}^n$ is continuous, then exists $x_0\in\mathbb{S}^n$ such that $f(x_0)=f(-x_0)$) can be sucesfully proved by using the Brouwer ...
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0answers
62 views

$\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ [closed]

I was stuck by reading this figure: It looks that $\mathbb{T}^4/\mathbb{Z}_2$, $\mathbb{T}^2/\mathbb{Z}_2$, and $\mathbb{CP}^1$ are somehow related. Are there some easier explanations from math ...
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Problem 1.3.2 in Guillemin and Pollack.

The question and its answer is given below: But I could not understand 1- why he defined $\psi(\tilde{U}) = U$? I know that the importance of $\tilde{U}$ to adjust the dimension. 2-why $\psi ^...
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3answers
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Surjectivity of a continuous map implies surjectivity on $\pi_1$

Let $X\to Y$ be a continuous surjective map between path-connected compact topological spaces (say, CW complexes), such that every fiber is path-connected. Can it be true that it always induces the ...
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1answer
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why it is called “extension of partial converse 2” GP1.4.4?

The partial converse 2 is given below: And the question is given below: 1-I am wondering why this problem considered as an extension of partial converse 2? could anyone answer this question ...
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1answer
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Small perturbation of $f$ doesn't affect the topology of $f^{-1}(0)$

This question arise from the proof of the degree-genus formula which asserts that a non-singular complex algebraic curve $C=\{x\in \mathbb{CP}^2| p(x) = 0\}$ is a (real) surface of genus $\frac 1 2(d-...
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1answer
95 views

Show that any cell complex is homotopy equivalent to a CW complex

We will give the definitions of cell complex and CW complex we use at the end of the post. Briefly speaking, "in a cell complex you don't have to glue cells in the order of their dimension, whereas ...
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1answer
46 views

A difficulty in the proof of partial converse I.(p. 24 in Guillemin & Pollack)

The statement and its proof is given below: But I am wondering: How can I check that 0 is a regular value for $g$?
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1answer
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Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds. (Guillemin & Pollack p.23)

Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds? $M(n)$ is the space of all $n x n$ matrices and $S(n)$ is the space of all $n x n$ symmetric matrices. ...
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1answer
39 views

A confusion in finding the tangent space of $O(n)$ group. [closed]

The question and the solution is given below: But I could not understand the last 2 paragraphs, why the kernel of $df_{A}$ is as described and why the dimension of the subspace is n choose 2, ...
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2answers
93 views

A confusion in the proof of Stack of records theorem.

The theorem and its Proof is given below: But I do not understand in the last line of the proof, why he said that $Z$ is closed and why he is sure that it does not contain $y$ ? The hint ...