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Questions tagged [algebraic-topology]

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

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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Homology groups of unit square with parts removed

I did exercise 19 in Hatcher on page 132 and I was wondering if anyone could tell me if this is right: 19. Compute the homology groups of the subspace of $I \times I$ consisting of the four boundary edges plus all points in the interior whose first…
Rudy the Reindeer
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Borsuk-Ulam theorem and two different pairs of antipodal points

Borsuk - Ulam theorem states in particular that for any continous map $f:S^2\rightarrow \mathbb{R}^{2}$, there exists point $x\in S^{2}$ such that $f(-x)=f(x)$. I call $(x,-x)$ a "good" pair My question is: How should $f$ look like if we want to…
mkultra
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What is 1st homology group of $X = \text{plane} - \bigcup_{n}\{(1/n, 0)\}\cup\{(0,0)\}$?

What is 1st homology group of $X = \text{plane} - \bigcup_{n}\{(1/n, 0)\}\cup\{(0,0)\}$? My guess is since it is $\operatorname{Ab}(\Pi_1(X))$. It is a subgroup of $\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}\dotsb$ countably many times, generated by…
rohit
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Number of antipodal pairs of $n$-sphere points mapped to discrete points in $\mathbb{R}^n$

I'm thinking on the Borsuk-Ulam theorem, which states that If $f\colon\mathbb{S}^n\to\mathbb{R}^n$ is continuous, then there exists an $x$ such that $f(x)=f(-x)$. This means that if $f$ is continuous, then at least one pair of antipodal points of…
nullgeppetto
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Calculating the Chern class of the normal bundle of $\mathbb{CP}^{n}$ inside of $\mathbb{CP^{n+1}}$

I found this bundle to be trivial. However I am having trouble to persuade one of my fellow graduate students that this is true. He reasoned as follows: 1) The inclusion map $i: \mathbb{CP}^{n}\rightarrow \mathbb{CP}^{n+1}$ give us a canonical…
Bombyx mori
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Singular homology of projective space

I am currently learning about homology theory. I have started with the definition of singular homology, and I am trying to compute the homology of some spaces we often use, but I am stuck with the real projective spaces. The result I want to get…
Van
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How can I show $\mathbb{TS^{5}}$ is not trivial?

I want to ask if anyone has a hint how to solve this problem via elementary differential topology methods not involving topological $K$-theory like $H$-spaces, $J$-homomorphism, etc. Even if we use $K$-theory then we know…
Bombyx mori
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Homology of connected sums

How do you compute the homology groups of $n$ connected sums $H_m(T^2\#T^2\#\dots \#T^2)$ where $T^2$ is the cross product of 2 circles? I know how to compute the homology groups of $T^2$ minus a point and that I need to use the Vietoris-Mayer long…
gary
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Using Mayer-Vietoris sequence to show the Möbius band does not embed in $S^2$

I'm studying for an exam and one of the questions is to show that the Möbius band $M$ does not embed into $S^2$. Note that we have $M=[0,1]\times [0,1]/\sim$ where $(0,s)\sim(1,1-s)$ We were given the following hint: Suppose that $f:M\rightarrow…
samlanader
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Artin Algebraic Topology

I am reading algebraic topology by Artin, and he gives the following definition. Let $X$ be a topological space. We define a homomorphism $sd_X: S_p(X) \to S_p(X)$ by induction. If $T: \Delta_0 \to X$ is a singular 0-simplex, we define $sd_XT =…
user68330
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$RP^n\times S^m$ and $RP^m\times S^n$ not homotopy equivalent ($n\not=m)$

I'm trying to see why $RP^n\times S^m$ and $RP^m\times S^n$ are not homotopy equivalent. The tricky part is that they have the same homotopy groups(if n,m>1). I have spent countless hours on this question without any luck, I know that if $n
Simplyorange
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Showing that $\pi_n(X \vee Y) \cong \pi_n(X) \oplus \pi_n(Y) \oplus \pi_{n+1}(X \times Y, X \vee Y)$ for $n \geq 2$

(This problem is from chapter 9 of J. P. May's A Concise Course in Algebraic Topology.) At first I thought of showing that the exact sequence $$ \cdots \to \pi_{n+1}(X \times Y) \to \pi_{n+1}(X \times Y, X \vee Y) \overset{\partial}{\to} \pi_n(X…
ho boon suan
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Question about Relative Homology

I am reading through Hatcher and I came across the following statement and am having trouble making sense of it. I am not sure why elements may be written this way. Any help will be appreciated. By considering the definition of the relative…
dinky
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Covering space of figure 8 corresponding to $\mathbb{Z}$

Provided that certain conditions are satisfied, we know that there's a one to one correspondence between covers of a space and subgroups of the fundamental group of that space. Since $\mathbb{Z}$ is a subgroup of $F(2)$, the free group on two…
user67200
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$H_p(\mathbb{R}P^3 \times \mathbb{R}P^2)$

I'm working through an example of the Kunneth formula in my book. Without showing any working it states that for $X = \mathbb{R}P^3 \times \mathbb{R}P^2$ $$H_p(X)=\begin{cases} \mathbb{Z} & \mbox{if } p=0\\ \mathbb{Z}/2\mathbb{Z} \oplus…
Juan S
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