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

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2
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1answer
267 views

mapping cone and cylinder

Given a map of spaces $f:X \to Y$, the mapping cylinder is the adjunction space $$cyl(f)=(X \times [0,1]) \cup_f Y$$ where we regard $f$ as a map $f: X \times \{1\} \to Y$.\ On the other hand the ...
1
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0answers
74 views

Correspondence between first homology group and deck transformations.

Let $M$ be a connected topological manifold with universal covering $\pi: \widetilde{M} \rightarrow M$ and let $p \in \widetilde{M}$ be a point. Let $\alpha,\beta : \widetilde{M} \rightarrow \...
0
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1answer
99 views

Homotopy Invariance: Cone Construction and Prisms Operators

I'm looking at different approaches to proving the homotopy invariance of homology. Rotman and Dieck both mention "the cone construction", but hatcher only introduces the prism operators and does not ...
0
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0answers
42 views

Does the classifying map of a fibre bundle only depend on the transition functions?

Does the classifying map of a fibre bundle only depend on the transition functions? Precisely, Let $\xi$ and $\eta$ be two fibre bundles over $B$, whose transition functions are same, both of their ...
3
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1answer
116 views

Zero homology and path-component

I try to show the two next things : $H_{0}(X,A)=0$ iff $A$ meets each path-component of $X$ and $H_{1}(X,A)=0$ iff $H_{1}(A)\rightarrow H_{1}(X)$ is surjective and each path-component of $X$ ...
0
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0answers
69 views

What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent?

Suppose that X is a topological space and $x_0$, $x_1$ are points of $X$. What is meant by saying that two paths in $X$ from $x_0$ to $x_1$ are equivalent? I presume it's enough to just say the path ...
2
votes
1answer
47 views

Why a convergent succesion does not have the same homotopy type of a CW-Complex?

The question is pretty much in the title; If my space is $\{1/n\}_{n\in \mathbb{N}} \cup \{0\} $ why it isn't homotopically equivalent to a CW-Complex?
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2answers
330 views

Retract and homology

I have this problem in Hatcher's book : Show that if $A$ is a retract of $X$ then the map $H_{n}(A)\rightarrow H_{n}(X)$ induced by the inclusion $A\subset X$ is injective. I think I have ...
1
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2answers
94 views

Excision theorem

We consider $f:S^{n}\rightarrow S^{n}$ a continuous function. Then we consider $y\in S^{n}$ such as $f^{-1}(y)=\{x_{1},..., x_{p}\}$, $U_{1},..., U_{m}$ $x_{i}$-neighbourhoods, and $V$ y-neighbourhood ...
1
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1answer
64 views

$\mathbb RP^n$ as CW-complex

In hatcher (p.6) it says that $\mathbb RP^n$ can be obtained from $\mathbb RP^{n-1}$ by attaching an $n$-cell via the quotient map $S^{n-1} \to \mathbb RP^{n-1}$. I was wondering what this is for $n=0$...
1
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2answers
67 views

Manifold with $\pi_1(M)=F_n$

We may construct a 3-manifold $M_n$ with $\pi_1(M_n)\cong F_n$ (i.e. the free group on $n$ generators) as follows: consider the complement of $n$ pairs of open 3-balls in $\mathbb{R}^3$. For each pair,...
0
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1answer
87 views

Show that $\mathbb{R} P^3$ is not homotopy equivalent to $\mathbb{R} P^2 \vee S^3$.

I'm studying for an oral qualifying exam in algebraic topology, going through questions in various tests published on the interwebs. Here's a rather straightforward question from this exam that is ...
4
votes
2answers
138 views

Homomorphism between homotopy groups of spheres induced by the fibration and the multiplication map of $SO(n)$

Let $n$ be a nonnegative integer and $x\in S^n$ a point in the n-sphere. Combining the map $\alpha\colon SO_{n+1}\longrightarrow S^n$ induced by matrix multiplication with $x$ and the connecting ...
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0answers
51 views

Is the question in the Munkres's topology book wrong?

At the end of cheapter $8.1$, $4)$ Given spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. $b)$ Show that if $Y$ is path connected, the set $[I,Y]$ has a ...
0
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2answers
232 views

Hatcher problem 1.2.12

In this problem a modified Klein bottle (say $X$) is taken in account which is seen as embedded space in $\mathbb R^3$ (giving subspace topology on the usual self intersecting figure of Klein bottle ...
0
votes
3answers
36 views

$S^{n+m+1}$ can be decomposed as the union of $S^n\times D^{m+1}$ and $D^{n+1}\times S^m$ along their boundaries.

Let $S^n$ denote the $n$-sphere and $D^n$ denote the $n$-disk (of course, $\partial D^{n+1}\cong S^n$). Then $S^n\times D^{m+1}$ and $D^{n+1}\times S^m$ both have boundary $S^n\times S^m$. The ...
0
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1answer
214 views

Proving a nowhere vanishing vector field on 2D manifold implies $TU\cong M\times S^1$

So, I am trying to solve the following problem. Suppose you have a nowhere zero smooth vector field on a 2 dimensional oriented compact manifold. Prove that the unit tangent bundle $TU$ is ...
0
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0answers
42 views

Non-isomorphism of topological line bundles on a Riemann surface, from first principles only

Although this question is in the same vein as my previous query, Isomorphisms (and non-isomorphisms) of holomorphic degree $1$ line bundles on $\mathbb{CP}^1$ and elliptic curves, it is nonetheless ...
1
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1answer
97 views

Homology group of an infinite wedge product

I'm struggling with the following algebraic topology problem: I'm given a collection of (topological) spaces $(X_i)_i$, ($i\in I$ whatever) and for each space a point $x_i\in X_i$. Then $\bigvee_i ...
3
votes
1answer
121 views

Moduli spaces, stacks and homotopy theory

For my final-year project (not this academic year but next) I'm hoping to write a relatively complete account of the basic theory of schemes used in modern algebraic geometry. My supervisor thinks ...
2
votes
2answers
117 views

Compact surface homotopy equivalent to $\mathbb{CP}^2 \setminus \{p\}$

Let $p$ be a point $\in \mathbb{CP}^2$. Is there a compact surface which is homotopy equivalent to $\mathbb{CP}^2 \setminus \{p\}$ ? I know the homology of $\mathbb{CP}^2$, but I'm not sure about the ...
3
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2answers
125 views

Does there always exist a lift of a path from $Y$ to $X$ if $f: X\to Y$ is a continuous surjective function?

If one has a continuous surjective function $f:X \longrightarrow Y$ and let $\gamma$ be a continuous path in $Y$, under what circumstances can one find a (possibly non-unique) lifted path $\gamma'$ in ...
1
vote
1answer
60 views

Relation between cup product and intersection number

Suppose $M$ is an oriented diff. manifold and $X$ and $Y$ are two submanifolds of codimension $m$ and $n$ in $M$. Then under some conditions one can define the intersection of $X$ and $Y$ and this is ...
0
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0answers
102 views

deformation retraction for a simple Hopf link

I would like to find a deformation retraction $F:X\times[0,1]\to X$ of the complement of the Hopf link which consists of two circles linked together once. Since Hatcher describes a deformation ...
2
votes
0answers
71 views

cohomology of complex grassmannians and quaternion grassmannians (the finite dimensional case)

Let $G_k(\mathbb{C}^N)$ be the complex grassmannian manifold. Let $G_k(\mathbb{H}^N)$ be the quaternion grassmannian manifold. Let $p$ be a prime integer (we may also impose extra conditions, such as ...
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0answers
223 views

Need help on how to compute the fundamental group of a space.

I'm studying for an oral qualifying exam and going through various past exams I find on the interwebs, including this mock exam from the University of Bath. One of the questions seems like it should ...
1
vote
1answer
63 views

What does $d_{n+1}\circ{d_{n}}=0$ mean in the definition of a chain complex?

According to the Wiki article on chain complexes, a chain complex $(A_{\bullet},d_{\bullet})$ is a sequence of abelian groups or modules connected by homomorphisms such that $d_{n+1}\circ{d_{n}}=0$. ...
2
votes
1answer
116 views

tensor product of trivial line bundles

In Hatcher's book on Vector Bundles, the tensor product of two vector bundles is defined through the gluing functions. But I need an example to understand it. So I think of the simplest case, the ...
2
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0answers
229 views

Mobius band does not retract to boundary circle - specific part

(I'm trying to ask this in a way such that it isn't a duplicate question) The proofs that I've seen for the fact that there is no retraction from the Mobius band to its boundary circle usually say ...
1
vote
1answer
98 views

Smash product and tensor product of groups

The smash product acts like a 'tensor product' in the category of pointed spaces (i.e. when the spaces are locally compact Hausdorff smashing is associative and satisfies a tensor-hom adjunction). ...
2
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0answers
57 views

Proving $S^{4}/G$ is simply connected where $G$ is not a free group action

Consider the sphere $S^{4}$ as a subset of $\mathbb{R}^{5}$ and consider the action of the group $G$ of homeomorphisms generated by $(x_1, x_2, x_3, x_4, x_5) \rightarrow (-x_2, x_1, -x_4, x_3, x_5)$ ...
0
votes
2answers
44 views

$A \xrightarrow{\alpha, \beta} E \xrightarrow{p} B$, where $p$ is a covering map and $A$ is connected.

Problem: Let $\alpha, \beta$ be continuous maps from a connected space $A$ to a space $E$, and $p:E \rightarrow B$ a covering map. If $a \in A$ with $\alpha(a) = \beta(a)$ and $p \circ \alpha = p \...
1
vote
1answer
85 views

Absolute extensors of $\mathbb{R}$

I want to prove that the only absolute extensors of $\mathbb{R}$ are the intervals. To prove that any interval is an absolute extensor I do: For any closed interval $[a,b]$ I apply Tietze Theorem. ...
2
votes
1answer
89 views

Attaching cells gives isomorphism of homotopy groups

I want to prove the following statement: Let $(X, x_0)$ be a pointed space, and let $X' = X\cup_{\alpha} e^{n+1}$ be obtained from $X$ by adjoining an $(n + 1)$-cell. Then the inclusion $i : X\...
2
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0answers
100 views

Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
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0answers
39 views

Dual of path object

For a topological space $X$, what might be the dual of the path space $X^I$ of $X$? Does it make any sense to think of the topological cylinder $X \times I$ over $X$ as dual to the path space over $X$?...
2
votes
2answers
90 views

Fibration $p_1:P(Y,y_0)\to Y$ has section iff $Y$ is contractible.

Let $P(Y,y_0) = \{ \omega : \omega(0) = y_0 \}$ be path space let's consider a fibration $p_1:P(Y,y_0)\to Y$ such that $\omega \mapsto \omega(1)$. Show that there exists $s: Y \to P(Y,y_0)$ such that $...
1
vote
1answer
97 views

Killing homotopy groups

I am basic with homotopy theory and especially with CW-approximation, Postnikov and Whitehead towers. In the proof of such things one need the following result on and on: Let $X'$ be obtained from $X$...
2
votes
1answer
380 views

Does there exist some relations between Cryptography and Algebraic Topology? [closed]

We know that there are many application of Cryptography in our real life. Are there any relation between Cryptography and Algebraic Topology? If yes, please suggest me some link or books. Thanks ...
0
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1answer
40 views

The group of continuous homomorphisms

Consider the topological group $\Bbb Q$ with the subspace topology and $\Bbb Z$ with the discrete topology. Is there a characterization for $C^{1}(\Bbb Q,\Bbb Z)$, the group of continuous ...
4
votes
1answer
186 views

Understanding de Rham cohomology: geometrically speaking, when is a smooth function closed

On Wikipedia the de Rham cohomology groups are defined to be the cohomology groups of the de Rham cochain complex (equivalence classes of differential $k$-forms). By this definition the zeroth de ...
4
votes
1answer
156 views

Finding the degrees of the attaching map of the $2$-cell of the torus

I am trying to calculate the degrees of the attaching map of the two cell of the torus. I have the following cell structure: The $2$-cell is $e_2$ and the $0$-cell (all four corners) is $e_0$. I ...
7
votes
2answers
1k views

Book for Algebraic Topology- Spanier vs Tom Dieck

A number of times, questions have been asked on this website about good books on Algebraic Topology and the responses have been very valuable. However I need some more specific advice in this matter. ...
2
votes
1answer
82 views

Is this a covering space of $S^1 \vee S^1$?

Is the following a covering space of $S^1 \vee S^1$ ? It would appear so since there is no point that has more than 2 incoming or outgoing arrows. It seems that the potential covering map $p:Y\to S^...
2
votes
0answers
30 views

CW approximation of $n$-connected space

I want to prove the following lemma: Let $X$ be a n-connected space. Then there exists a CW-approximation $f:K\rightarrow X$ such that $K$ has trivial n-skeleton. What I have done so far: By ...
3
votes
0answers
241 views

Fundamental group of connected sum for non-orientable manifolds

For orientable $n$-manifolds, $n\ge3$, the fundamental group of connected sum is free product of fundamental groups: $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$. As far as I understand, for non-orientable ...
3
votes
1answer
211 views

Obtaining the Möbius strip as a quotient of $S^1\times[-1,1]$

I am trying to obtain the Möbius strip as a quotient of $S^1\times[-1,1]$, where $S^1$ is, of course, the circle. My definition of Möbius strip is the quotient of the square $[0,1]\times[0,1]$ by the ...
1
vote
1answer
124 views

Surjection of the fundamental group of a manifold onto a free group induces a map onto a wedge of circles, why?

Why does a surjective map $\pi_1(M)\twoheadrightarrow F$ of the fundamental group of a manifold $M$ onto a free group $F$ over $n$ generators induce a continuous map $M\twoheadrightarrow\bigvee^n S^1$ ...
1
vote
1answer
42 views

The Projection $S^{2}\times S^{3} \rightarrow (S^{2}\times S^{3})/(S^{2} \lor S^{3})$

So a problem I have come across involves constructing some smooth degree 1 maps, and one such map to be constructed is a map $S^{2}\times S^{3} \rightarrow S^{5}$. I've been told that there is such a ...
1
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2answers
75 views

$f(x):S^{2n} \rightarrow S^{2n}$ continuous so that there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$

Let $f:S^{2n}\rightarrow S^{2n}$ continuous. Then there is $x \in S^{2n}$ with $f(x)=x$ or $f(x)=-x$. I am having a hard time finding a starting Point. Thank you