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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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21356 questions
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Continuous Map between different dimension Real Projective spaces

I've Been trying to prove for many hours now that given a (continuous) map $f:RP^n\to RP^m$ (where $n>m>0$) the induced map $f_* : \pi_1(RP^n)\to \pi_1(RP^m)$ is trivial. I've seen a few descriptions for real project spaces, the one I have been…
MrP
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Fundamental group isomorphic to zero element

Show that $\pi_1(X,x_0)\cong 0$ if and only if for any loop $f:[0,1]\rightarrow X$, $f(0)=f(1)=x_0$, there exists a continuous map $g:[0,1]\times[0,1]\rightarrow X$, such that $g(0,s)=f(\frac{s}{2})$, $g(1,s)=f(1-\frac{s}{2})$, $g(t,0)=x_0$,…
Nothing
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topology on CW-Complex $K$ or $n$-skeleton $K^{(n)}$

I have a hard time understanding what the topology on CW-complexes is, i.e. what the open sets of a given CW-Complex $K$ or $n$-skeleton $K^{(n)}$ are. From my understanding we get the topology inductively and we start with the discrete topology on…
Ton910
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Only finite group that acts on even spheres is $\mathbb Z_2$

If $G$ is acting freely on $\mathbb S^{2n}$ we can associate to $\rho\in G, \rho\neq0$ it's degree $deg\rho=(-1)^{n+1}$ as any map without fixed points has that degree. This shows that every non-trivial element is homotopic to the antipodal map. Why…
Aner
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the boundary of Mobius band is homeomorphic to a circle

It is know that $\partial M\approx S^1$, where $\partial M$ is the boundary of the mobius band. But how do we prove that? I am trying to define the homeomorphism explicitly, but it is messy. The Mobius band $M$ is defined by the quotient $q:I\times…
user854186
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exact homology sequence associated to a fibration

let $p:E\rightarrow B$ be a fibration with fiber $F$, ($E$ and $B$ are cw complexes). $B^k$ denotes the $k$-skeleton of $B$. 1) what does this sentence mean :"we denote the restriction of $E$ to $B^k$ by $E^k$". 2) why there is an exact…
firas
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Does the sphere $S^3$ nontrivially cover itself?

I am having a hard time deciding whether or not $S^3$ nontrivially covers itself. Some help would be appreciated. Thanks
caley
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Pseudo-Anosov mappings

Am working on a problem for my grad math class, but I having some trouble intuitively understanding the idea of pseudo-Anosov mappings. I am trying to do something involving $a^{-1}b^{-1}ab$, where $a^{-1}b^{-1}ab=c$ for a pseudo-Anosov mapping $c$,…
Anonymous
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What is the reduced suspension of $I=[0,1]$?

How to visualise the reduced suspension of $I=[0,1]?$ Here is an answer, Smash product of $S^1$ with the interval $I$ but I don't get it. So, it will be helpful for me if you can give a pictorial view.Thanks
SOUL
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How to compute the homology group $H_q(\mathbb{R}^n-e^r)$?

Let $e^r$ be a homeomorphic copy of $I^r$ in $\mathbb{R}^n$($I=[0,1]$).How to compute the homology group $H_q(\mathbb{R}^n-e^r)$?($r,n,q$ are non-negative integers)
Nirvanacs
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Small confusing regarding real projective line

In exercise 8.2 of Rotmans "An Introduction to Algebraic Topology", we have to show $\mathbb{R}P^1\cong S^1$, and in exercise 8.5 that for all $n\geq0$ we have $\mathbb{R}P^n\cong S^n/\sim$ where $\sim$ identifies antipodal points. So, does this…
Thanks.
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Hatcher exercise 0.6(c)-continuity

I want to show the continuity of the function defined in here. But I don't know how to. Any hints or comments will be appreciated.
one potato two potato
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Why are Mapping Cylinders of “nice” functions CW Complexes?

I keep running across comments and answers to questions that imply that the mapping cylinder for “nice” functions is a CW complex. Why is this necessarily so? Consider any function $ f \colon X \to Y $. Using Hatcher’s (p2) standard definition,…
PossumP
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Naturality of the cross product

Apologies for the vague title, but I can't describe the following question well. The question is (as usual, from my book) Let $f:X \to X'$ and $g:Y \to Y'$ be continuous. If $\mbox{cls} \varphi \in H^p(X';R)$ and $\mbox{cls} \theta \in H^q(Y';R)$,…
Juan S
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Algebraic Topology Hatcher Chapter 3.2 Problem 3

The Problem: Using the cup product structure, show there is no map $\mathbb{R}P^n \rightarrow \mathbb{R}P^m$ inducing a nontrivial map $H^1(\mathbb{R}P^m; \mathbb{Z_2}) \rightarrow H^1(\mathbb{R}P^n; \mathbb{Z_2})$ if $n > m$. What is the…
Math_Day
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