Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

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1answer
38 views

union of two contractible spaces, having nonempty path-connected intersection, need not be contractible

show that union of two contractible spaces, having nonempty path-connected intersection, need not be contractible. can someone give me a proper example please.I could not remind anything.
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0answers
165 views

When is Quotient Map a Covering Map

Group $G$ acts on topological space $X$. Also, $x,x'\in X$ not in the same orbit of $G$ have open $U$, $U'$ such that $g(U)\cap U'=\varnothing$ for all $g\in G$. I have shown that $X/G$ is ...
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1answer
28 views

Orbifolds and singular points

If I understand correctly, we roughly define the singular locus $\Sigma_O$ of an orbifold $O$ to be the set of points where the orbifold fails to be a manifold. In particular, if $x \in O$ has a ...
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0answers
21 views

Show, that in any topology space $X$, if $A\subseteq X $ then $ Cl_{X}(Cl_{X}(A))=Cl_{X}(A)$ [duplicate]

Show, that in any topology space $X$, if $A\subseteq X $ then $ Cl_{X}(Cl_{X}(A))=Cl_{X}(A)$
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1answer
82 views

relation between the group O(3) and SU(2)

Base on relations between groups $O(3)$, $SO(3)$ and $SU(2)$: a) $O(3)=SO(3)\otimes \{1,-1\}$ b) $SO(3)\simeq SU(2)/Z_2$ Can I say $\{1,-1\}$, i.e. $Z_2$, also the center of the group $O(3)$? If ...
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2answers
68 views

Computing the fundamental group with Seifert-van-Kampen

I need some help to solve this: Let $X := \Bbb R^{2} \setminus \lbrace x_{0},x_{1}\rbrace$, where $x_{0}$ $\not= x_{1}$ and $x_{0},x_{1} \in \Bbb R^{2} \setminus \lbrace \left(0,0\right)\rbrace$. ...
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0answers
59 views

Question on Singular homology

i have this example : The homology of the space $X=\lbrace x \rbrace$ . for all $p\geq 0$, there is a unique singular p-simplex $\sigma_p:\Delta_p\rightarrow X$, and for $p>0$ we have ...
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0answers
49 views

Borsuk Graph and chromatic number

For a positive real number $\alpha < 2$, let $B(n+1,\alpha)$ be the (infinite) Borsuk graph with $S^n$ as the vertex set and with two points connected by an edge iff their distance is at least ...
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2answers
106 views

If all paths with the same endpoints are homotopic, then the space is simply connected.

Let $X$ be a path connected space such that any two paths in $X$ having the same end points are path homotopic. Then prove that $X$ is simply connected. I am totally stuck on this problem. Can ...
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0answers
67 views

On the Proof of the Perron-Frobenius Theorem.

The Perron-Frobenius theorem states that a square matrix with nonnegative entries has a real nonnegative eigenvalue. One possible proof uses the Brouwer fixed point theorem, and every proof I've seen ...
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1answer
115 views

CW-Complex: Characterisation of open sets

I want to prove the following little Lemma: Let K be a CW-complex and let $S \subset K $. Prove that: S is open in K if and only if $f_{\sigma}^{-1}(S)$ is open for each $\sigma$. Where: ...
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1answer
52 views

Mayer-Vietoris for a cover without triple intersections

Let $M = \bigcup_i U_i$ be a cover with open sets $U_i$ such that for for distinct $i,j,k$ we always have $U_i \cap U_j \cap U_k = \emptyset$. I would like to show the existence of the following ...
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1answer
131 views

Mapping cone not homotopy equivalent to quotient space

My question is about a "non-example" to theorem 1.6 in chapter VII in Bredon. We have an inclusion $i: A \to X$, with $A = \{0\} \cup \{1/n | n = 1,2,...\}$, and $X = [0,1]$. Then $X/A$ is a ...
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3answers
221 views

Path connected spaces with same homotopy type have isomorphic fundamental groups

I was try to understand the following theorem:- Let $X,Y$ be two path connected spaces which are of the same homotopy type.Then their fundamental groups are isomorphic. Proof: The fundamental groups ...
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0answers
58 views

Different versions of Hatcher

I suddenly found out that my Hatcher from amazon is very different from the version on his website. Should I assume his website is up to date, and hence my copy is an old version?
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3answers
68 views

what is the meaning of a symbol $\pi_1(X,x_0)=0$

what is the meaning of a symbol $\pi_1(X,x_0)=0$ when $X$ is a contractible space to $x_0$. Actually I know that $\pi_1(X,x_0)$ is the fundamental group of $X$ based at $x_0$.But I could not ...
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2answers
54 views

Some questions on the definition of $n$-simplex.

Hatcher P102 last paragraph: $n$-simplex is the smallest convex set in $\mathbb{R}^m$ containing $n+1$ points $v_0, \dots, v_n$ that do not lie in a hyperplane of dimension less than $n$. I ...
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1answer
37 views

$a - b, b-c, c-d$ form a basis for this kernel.

Hatcher P99 last paragraph: Define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edge minus the vertex at the tail. Thus ...
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12answers
1k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
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0answers
45 views

Serre spectral sequence and locally constant coefficients

I have a brief question - In the Serre Spectral sequence for a fibration $$F \rightarrow E \rightarrow B$$ one can require, to avoid using local system of coefficients, that the action $\pi_1(B)$ on ...
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1answer
33 views

Topological interpretation of a zero map.

I have a pair of complexes of modules $A_{\bullet}$ and $B_{\bullet}$, and I want to create a new complex pretty trivially by shifting one complex by 1, and then taking the direct sum of the ...
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1answer
55 views

Modification of Excision Theorem

In the Excision Theorem, there is a condition that the closure of U is contained in the interior of A. Now I wonder if the Excision Theorem is still true when this condition is replaced by the ...
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1answer
10 views

Sending each basis element $a,b,c,d$ to $y-x$.

Hatcher P99 last paragraph: Define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edge minus the vertex at the tail. So I ...
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1answer
215 views

Fundamental Polygon of Real Projective Plane

Wikipedia gives the following fundamental polygon for the real projective plane $\mathbb{R}\mathrm{P}^2$ The problem here is that the corners aren't identified to a single point (like in the ...
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1answer
92 views

to understand a theorem for fundamental group

I faced a problem to understand the proof of the following theorem from the book "algebraic topology by satya deo". If $F\colon X\to Y$ be a homotopy between two maps $ f,g\colon X\to Y $. Let ...
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5answers
96 views

Free product of the trivial group with another group

I'm new to the idea of a free product.. Basically I was wondering if G is an arbitrary group and 1 is the trivial group then is $1\star G \cong G$. If not.. what whould it look like?
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1answer
186 views

How to show the standard $n$-simplex is homeomorphic to the $n$-ball

I am trying to show the standard $n$-simplex is homeomorphic to the $n$-ball. Here, the standard $n$-simplex is given by $$\Delta^n=\left\{(x_0,x_1,\cdots,x_n)\in\mathbb{R}^{n+1}:\sum ...
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2answers
85 views

Help with constructing homeomorphism for this identification

Consider the following triangles: I have shown that $T$ is homeomorphic to as disc. Here is the proof: First note that one can prove the following theorem: The mapping $f^\ast : X/\sim_f \to Y$ ...
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1answer
100 views

Retraction of surface of genus $g$

This is an exercise in 53 page of Hatcher's book : Consider surface of $M_g$ of genus $g$ If $g=h+k$ then $$M_g = M_h'\cup_{S^1} M_k'$$ where $M_h' = M_h - D^2$ Question 1 : Then show that $M_h'$ ...
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2answers
161 views

Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta} $ where $d_{\alpha\beta}$ is ...
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0answers
167 views

Galois correspondence of covering spaces of spaces not necessarily semilocally simply-connected

I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology: Given a covering space action of a group $G$ on a path-connected, locally path-connected space $X$, ...
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0answers
41 views

Why Is the Induced Map Not Zero?

I am reading "Modern Classical Homotopy Theory" by Strom and have come across the following. We are given a fibration $F\rightarrow E\rightarrow B$. One then has two pushout squares: ...
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1answer
152 views

The Stiefel-Whitney classes of Cartesian product

I am reading the book of characteristic classes by Milnor-Stasheff, and I have a problem with the exercise 4-A: Show that the Stiefel-Whitney classes of a Cartesian product are given by ...
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1answer
48 views

The Hopf invarient with coefficients other than Z.

So generally one defines the Hopf invariant of a map $f: S^{2n-1} \to S^n$ as the coefficient $H(f)$ in $\alpha^2 = H(f) \beta$ where $\langle \alpha \rangle = H^n(C_f)$ and $\langle \beta \rangle = ...
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1answer
158 views

Surgery on trivial knots

I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot. I think I read or heard somewhere that as a surgery link, we can take ...
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1answer
71 views

Question on functors

please i need help,how to prove that "the functor (covariant) "fundamental group", of the category of pointed topological spaces in the category of groups" is really a functor What i must do to ...
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1answer
37 views

Is this also a homotopy

If $F$ is a path at a point $x$ then the following defines a homotopy from the path $FF^{-1}$ to the constant path $e$: $$ \begin{array}{cc} H(t,s) = F(2t) & s \ge 2t \\ H(t,s) = F(s) & s \le ...
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1answer
71 views

Map induced by $O(n)\hookrightarrow U(n)$ on homotopy groups

There is an inclusion $O(n)\hookrightarrow U(n)$ which views an $n\times n$ orthogonal matrix as a unitary matrix. It is also a theorem, sometimes called Bott periodicity, that we have the following ...
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0answers
26 views

Factoring a map into a pointed acyclic cofibration

I am reading through Strom's paper "The homotopy Category is a Homotopy category" to better understand some exercises in "Modern Classical Homotopy Theory" and I am having some trouble with one thing: ...
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0answers
101 views

Homology groups (?) of some quotient of $\Bbb{R}P^n$

Here is a question from Hatcher (2.2.19): I assume that those $H_i$'s are homology groups, Hatcher denotes both the chain complexes and the homology groups by $H$ (I will denote the chain complexes ...
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1answer
80 views

Simply connected subset of $\mathbb R^3$

Let $C$ be the closed unit cube in $\mathbb R^3$, and let $A$ be one face of the cube $C$ (say the face above and parallel to $xy$-plane). Let $U\subset\mathbb R^2$ be open and path-connected such ...
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0answers
35 views

Finding a homotopy map

Let $K=\mathbb R^2\times (-\infty,0)\subset \mathbb R^3$, and let $Q$ be an open connected subset of $\mathbb R^2$. Is the fundamental group $\pi_1(Q\times [0,1)\cup K)$ trivial? And is it possible ...
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0answers
68 views

Question About Transgression

I have been working on this question here. Here is the setup: First, all cohomology groups are assume to be with $\mathbb{Q}$ coefficients. We assume that $H^*(K(\mathbb{Q},n))=\mathbb{Q}[x]$, with ...
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1answer
60 views

More interesting examples of spaces that are retractions?

I learned about retraction: A continuous map $r: X \to A$ where $A$ is a subspace of $X$ is called a retraction if $r|_A = id_A$. I made some examples. For example: If $D$ is the closed unit disk and ...
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2answers
203 views

Function doesn't have a lift in a space related to Topologist's sine curve

I'm trying to solve exercise 1.3.7 in Hatcher's Algebraic Topology: Let $Y$ be the quasi-circle that is the union of a portion of the graph $y = \sin(1/x)$, the line segment $[-1,1]$ in the ...
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0answers
150 views

Homology of nonorientable surfaces

Let $N_g$ be a closed nonorientable surface of genus $g$. I will try to compute the homology groups and I want you to help me with certain steps and correct my mistakes - I will use this as an ...
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0answers
32 views

Rational Elienberg-Maclane Spaces

Is it true that $$ H^k(K(\mathbb{Z},n);\mathbb{Q})\cong H^k(K(\mathbb{Q},n);\mathbb{Q}) $$ for all $k$?
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43 views

Seemingly serious problem with Deformation Retraction

So a friend and I are arguing over Deformation retraction. Any help to settle this would be nice. Consider a T shaped subspace of $\mathbb{R}^2$. Let $A$ be the vertical segment and let $B$ be the ...
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31 views

fibre bundle on [a,b]. prove that every fibre bundle on it is trivial.

prove that every fiber bundle on [a,b] is trivial. Please prove this elaborately. Is the Lebesgue covering lemma is required to prove this result?
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40 views

Collapse of a subspace - Cofibration

Let $i:A \rightarrow X$ be a (closed ) cofibration (i.e a cofibration in the Strøm Model structure). For a subspace $B \subset A \subset X$, when is it true that $A/B \rightarrow X / B$ is a ...