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

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0
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0answers
24 views

vector bundles of $\mathbb{P}^2$ [closed]

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
0
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1answer
38 views

Postnikov tower of a product

Let $X$ and $Y$ be simply connected, locally finite CW-complexes and let $(X_i)_i$ and $(Y_i)_i$ be their Postnikov towers respectively. Is the Postnikov tower of $X\times Y$ given by the products ...
2
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1answer
56 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
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0answers
90 views

Why every fundamental group isn't a trivial fundamental group?

I don't understand exactly the definition of the Fundamental Group. Munkres defined Fundamental Group in your book 'Topology' like "Let $X$ be a space; let $x_0$ be a point of $X$. A path in $X$ that ...
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1answer
72 views

Cellular homology of the real projective space $\mathbb R P^n$

I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: ...
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1answer
103 views

How should I prove the following? Algebraic topology and homeomorphism

I am struggling immensely with topology since the start of the course, probably due to its extremity; the explanations are either "very rough" or "very strict and rigid and hard to comprehend." Either ...
0
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1answer
90 views

What does the notation “*” mean?

I do not know the name of or what it does so I have no means of searching for an answer over the internet or a book. In my notes for algebraic topology, I have this bit that says, For any $f: X ...
0
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0answers
32 views

$K(G,1)$ for a torsion-free group $G$

It is known that if a finite CW complex $X$ is a $K(G,1)$, then the group $G$ must be torsion-free. see proposition 2.45 of Hatcher Now my question is If $G$ is a torsion-free group, then is ...
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0answers
29 views

Existence of a curve with index 1 around a compact set

Let $K \subset \mathbb{C}$ be compact. If $U$ is an open set containing $K$, I want to show that there exists a collection of (piecewise $C^1$) curves $\gamma_1...\gamma_n$ such that 1) For $ x \in ...
2
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0answers
23 views

Cohomology of geometric realization of a simplicial topological space

Let $X$ be a simplicial topological space. We can consider to notions of cohomology of $X$. Denote by $|X|$ geometric realization of $X$. Then we can take just $H^*(|X| )$ (i.e. usual cohomology of ...
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0answers
8 views

Filtration on simplicial complex as a grayscale image

If we make a filtration on the complex built from vertices and edges and faces(2dimensions) of a grayscale image according to grayscale value. Does it considered as a filtration on simplicial complex? ...
1
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1answer
49 views

$\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
1
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1answer
36 views

A remark about the map $\partial^*:H^{*}(A;V)\to H^{*+1}(X,A)$ of the l.e.S. in cohomology

My question is about a remark from lecture about the connecting-homomorphism of the long exact sequence in homology of a pair $(X,A)$. Let $(X,A)$ a pair of topological spaces, $V$ be an abelian ...
4
votes
1answer
50 views

Homotopy equivalence between $X=\{0\}\cup \{\frac{1}{n},n\in \mathbb{N}\}$ and a discrete space

Consider the space $X=\{0\}\cup \{\frac{1}{n},n\in \mathbb{N}\}$ with the topology induced by the real line. Is $X$ homotopy equivalent to some enumerable discrete space $Y$? My try was the ...
3
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1answer
45 views

counterexample for $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence

I want to understand a counterexample for: Let $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence, because the statement ...
4
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0answers
44 views

Characterisation of Stable Cohomological Operations: $\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$

I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological ...
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0answers
24 views

How do I prove shoenflies theorem for $\mathbb{R}^2$?

I studied the contents in Munkres-Topology. In this text, the author uses basic algebraic topology to prove Jordan curve theorem. Then, he wrote that "If $C$ is a simple closed curve in $S^2$, the ...
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1answer
46 views

Proof check/ suggestion: The suspension of $S^n$

In one of my excercise sheets there was a remark saying that $$SX \approx S^{n+1}$$ where $SX$ denotes the suspension of $X=S^n$. So I tried to prove this on my own and would like to discuss my ...
1
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1answer
47 views

Example of Spherical Element (Simplicial Homotopy)

Definition: An element $x\in X_n$ is said to be spherical if $d_i x=*$ for all $0\leq i\leq n$. $X$ is a pointed fibrant simplicial set. I am puzzling over this definition. For instance, if ...
1
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1answer
36 views

Degree of a non-surjective map f

In my notes I found an excercise claiming that $f: S^n \to S^n$ has $deg(f)=0$ whenever it's not surjective. I can prove this if I assume smoothness by applying Sard's theorem but I'm wondering if ...
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1answer
35 views

Can a torus be cut into a Möbius strip with zero number of half twists?

It is known that the torus can be cut into a Möbius strip with an even number $n$ of half twists(half twist means rotation 180 degree). I am asking if it is possible to $n$ to be zero?
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0answers
37 views

Topological Equivalence of Metric Spaces [closed]

Suppose we have two different metric spaces $(X,\phi)$ and $(Y,\psi)$. I need to show that the metrics $\phi$ and $\psi$ are equivalent metrics. Using a sterographic projection, I've shown that if we ...
7
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0answers
96 views

Codimension $1$ Embedding into $\mathbb{R}^{n+1}$

I am trying to determine which homotopy types can be realized by $n$-manifolds that have codimension one embeddings into $\mathbb{R}^{n+1}$. Suppose I have $X^n \subset \mathbb{R}^{n+1}$ and an ...
2
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0answers
49 views

Cohomology algebra generated by Steifel-Whitney classes and dual classes subject to defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
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1answer
35 views

Question about $C_f \approx \mathbb R P^2$

I'm trying to understand why the mapping cone $C_f$ (where $f: S^1 \to S^1, \space e^{2\pi it} \mapsto e^{4 \pi i t}$) is homeomorphic to the real projective space $\mathbb R P^2$. If we use the ...
0
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1answer
47 views

Question in the proof of the Brower fix point theorem

One can show that for any given homology theory $H$ with non-trivial coefficient group $G$ there does not exist a retract $\partial B^n \subset B^n$. Brower's fix point theorem states that any ...
2
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1answer
41 views

How do we obtain the following identification

I don't understand geometrically why the identification below let us generate the shape on the right can someone explain or give me some intuition ?
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0answers
34 views

Question about the Hessian Criterion on a curve with singularity

So in class we have this theorem we call the Hessian Criterion: If we have a singular point in an affine curve in $\mathbb{C}^2$. Then $\frac{ \partial ^2 f}{\partial x^2}\frac{ \partial ^2 ...
3
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0answers
39 views

Correspondence defines embedding of $G_n(\mathbb{R}^m)$ into $G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = G_{n+1}(\mathbb{R}^{m+1})$? [closed]

Does the correspondence$$X \overset{f}{\to} \mathbb{R}^1 \oplus X$$defines an embedding of the Grassmann manifold $G_n(\mathbb{R}^m)$ into $$G_{n+1}(\mathbb{R}^1 \oplus \mathbb{R}^m) = ...
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0answers
44 views

There is a theorem analogous to the Brouwer fixed-point for the 2-dimensional sphere?

Intuitively I think that for $f$: $\mathbb{S^2}\rightarrow\mathbb{S^2}$ a continuous function exist at least two fixed points. Using the same reasoning the statement of Brouwer fixed-point theorem I ...
2
votes
1answer
44 views

Given a column vector, can we add columns, continuously dependent on the given one, to get an invertible matrix?

Given a vector $x$ in the $n=6$-dimensional Euclidian space $\mathbb{R}^n$, do there exist $n-1$ continuous functions $f_1$ to $f_{n-1}$ such that the matrix $$(x,f_1(x), … ,f_{n-1}(x))$$ is ...
2
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1answer
38 views

How is identification is done in the definition of CW complexes

Consider the definition of CW complex from hatcher I am trying to understand the issue with the identification, because I feel there is something I don't understand. I decided to do an example and do ...
1
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1answer
25 views

Free group action on $S^n$ proof in Hatcher

Theorem: :$\mathbb{Z}_2$ is the only nontrivial group that can act freely on $S^n$ if $n$ is even. Proof: Since the degree of a homeomorphism must be $\pm 1$, an action of a group on $S^n$ determines ...
1
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0answers
47 views

Defining a continuous complex logarithm on open set $U \subset \mathbb{C}$

Suppose you are given an open set $U \subset \mathbb{C}$ and a continuous function $f: U \rightarrow \mathbb{C}-\{0\}$. And $f$ has the next property: For every closed loop $ c: I \rightarrow U$ ...
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0answers
53 views

Question about the induced Hurewicz isomorphism

In my notes it's claimed that the group homomorphism $$\Phi: \pi_{1}(X,x_{0}) \to H_{1}(X), \space \{f\} \mapsto[f]$$ clearly induces a group homomorphism $\Phi_{*}: \pi_{1}(X,x_{0})^{ab} \to ...
1
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0answers
51 views

Algebraic Topology problem on continuous functions over disks on sphere

I have come across the next problem, and I would like a little hint. Everything I'm thinking is not working or is a dead end. Let $f,g : D^2 \rightarrow S^2$ be continous maps such that $(x,y) \in ...
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0answers
31 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 ...
6
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0answers
50 views

Constructing $\mathbb{P^n}$ “bundle” with different $n$

Can we find a complex/smooth manifold and map $f\colon X\to\mathbb{C}^3$ such that it is a $\mathbb{P}_\mathbb{C}^m$ bundle over linear subspace $\mathbb{C}^1$ and $\mathbb{P}_\mathbb{C}^n$ bundle ...
0
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1answer
40 views

Show that Affine Curves are not compact

Hi guys I have a question and not sure how to connect the dots. I am suppose to show that over a algebraically closed field $K=\mathbb{C}$. The affine variety in $K \times K$ is never compact. There ...
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0answers
27 views

A question in Hatcher's proof of homotopy lifting property [duplicate]

In Hatcher's Algebraic Topology p.30, in the last but one paragraph, he said: After replacing N by a smaller neighborhood of $y_0$, we may assume that $\tilde{F}(N\times t_i)$ is contained in ...
3
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2answers
60 views

Elementary geometric characterization of spheres?

I've read the following two theorems. Theorem. A compact connected metric space whose points are cuts points with the exception of at most two is homeomorphic to the unit interval. Theorem. A ...
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1answer
42 views

Can we get a torus by identifying surface with removed disc and mobius strip?

If we take a surface and remove a disc, then identify this resulting circle with the boundary circle.. does this produce a torus?
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0answers
63 views

How much algebra one needs to study algebraic topology and homotopy theory?

I wonder how much algebra(group theory, abstract algebra, linear algebra, ring theory, field theory) is assumed as a prerequisite in most of the modern algebraic topology text. For example, these ...
3
votes
1answer
81 views

Homology of knot exterior in general manifold for not null-homologus knot

I was trying to figure out the homology of the knot complement when $K$ is not a rational null-homologous knot ($[K]\neq 0\in H_1(X_K,\mathbb Q)$). We then know by half-lives half dies that the ...
3
votes
1answer
62 views

Compatibility between Unreduced Suspension Iso and Reduced Suspension Iso

I need some clarifications on these two "basic" things because I realised I was using them carelessly and now I want to know once and for all the relation between the two. Let us assume working with ...
0
votes
2answers
60 views

Is $S^{\infty}$ contractible?

Recently I was reading this post: Unit sphere in $\mathbb{R}^\infty$ is contractible? Then a doubt came across to me: why I can't consider the linear homotopy $H:I\times S^{\infty}\to S^{\infty}$ ...
4
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1answer
70 views

Complement of a knot that *isn't* rationally null-homologous

Let $K$ be a knot in a closed, oriented 3-manifold $Y$. It is a standard fact that if $K$ is (at least rationally) null-homologous, then $H_1(Y-K;\mathbb{Z})$ is isomorphic to $H_1(Y;\mathbb{Z})\oplus ...
0
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1answer
57 views

What is the general structure of the complex curve $xy=y^2$?

How can you determine how a complex curve looks like in four dimensions, especially near singularities? In my example, the curve $xy=y^2$ consists of the lines $y=x$ and $y=0$ ($x,y$ complex). I think ...
3
votes
2answers
38 views

Trivial loop on the $1$-Skeleton

Following Hatcher's proof of Hurewicz Theorem (version of 1999) we arrive at the point that we must show that the loop in the picture, created following the path $0,1,2,3,1,0,1,3,0,3,2,0,2,1,0$, is ...
0
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0answers
21 views

Calculating fundamental groups using retracts

I want to know if my calculations are correct. I am trying to find the fundamental group of the figure $X_n$ in $\mathbb{R}^2$ obtained by drawing $n$ lines between two points (the lines only meet at ...