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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Is the restriction map of sections of a fiber bundle a Serre fibration?

Suppose that for a contractible space $A$ we are given a fiber bundle $p:E\to A$ and denote for $B\subset A$ by $E(B)$ the restricted bundle. I have good reason to believe that in this situation the restriction map $r:\Gamma^0(E)\to\Gamma^0(E(B))$…
Martin Worsek
  • 851
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Deformation Retraction to a point

I have a question from Hatcher's Algebraic topology Chapter 0, problem 6: "Let $X$ be the subspace of $\mathbb{R}^2$ consisting of the horizontal segment $[0,1]\times\{0\}$ together with the vertical segments $\{r\}\times[0,1-r]$ for $r$ a rational…
William Stagner
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Pullback of a good pair by a fibration

A pair $(X, A)$ is a good pair (as defined in Hatcher) if $A$ is a deformation retract of some neighborhood $N$ in $X$. Suppose $\pi : Y \to X$ be a fibration, and let $B = p^{-1}(A)$. Hatcher claims in his proof of Theorem 5.3 (see page 530 here)…
Frank
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Pontryagin-Thom Construction and Poincaré Duality

How is the Pontryagin-Thom Construction related to Poincaré Duality? These are two important ideas in topology which I understand separately and I've heard there is a link but I haven't found a reference for it.
Manuel
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A specific example of a CW complex and a few questions concerning it.

The question I am facing is this one: Construct a CW complex X with a 0-cell x(n) for each natural number $n \geq 0$ and a 1-cell $D_{n}^1, n \geq 1$ which is glued to $x(0)$ at one end and $x(n)$ at the other. For each natural number $n \geq 1$…
Relative0
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Understanding the proof of the Seifert-van Kampen theorem

Here is the bit of the proof I didn't really understand: Let $X=X_1\cup X_2$, where $X_1$, $X_2$, $X_1\cap X_2$ and $X$ are path-connected and $X_1, X_2$ are open subsets of $X$. Moreover, assume that the generator set $G_1$ with the relations $R_1$…
Xena
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Examples of topological spaces $X, Y$ with opposing continuous bijections $f: X \rightleftarrows Y: g$ with non-isomorphic $\pi_1$

The existence of continuous bijections $f: X\rightleftarrows Y: g$ doesn't necessarily imply that $X$ and $Y$ are homeomorphic, if $f$ and $g$ aren't inverses of each other. I'm interested in examples of pairs of spaces that have distinct $\pi_1$…
mattematician
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Misprint in Switzer's Algebraic Topology?

I am currently reading Switzer's book "Algebraic Topology: Homotopy and Homology". On page 50, the proof of 3.30 c), he claims that a certian composition is something I can't see how it possibly can be what he states. Let $\beta':S^1 \rightarrow I…
Helle Karlsson
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contractible fiber implies spaces have same cohomologies

Let $p:X\rightarrow Y$ be a fiber bundle with fiber $F$ that is contractible. Clearly $\pi_n(X)\cong \pi_n(Y)$ for all $n$. But why does it follows that $H^{*}(X)\cong H^{*}(Y)$? Is it because the iso $\pi_n(X)\cong \pi_n(Y)$ is induced by a map?
user1040289
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1 answer

How can I start Algebraic K-Theory?

Can you please tell me what are pr-requisite area to learn Algebraic K-Theory ? I studied Linear Algebra , Rings and Modules ,Fields and Galois Theory ,basic Commutative Algebra , Calculus on Differential Forms, Algebraic Topology without…
Math Learner
  • 51
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$X$ is a connected space and $Y$ is a discrete space prove that the two maps $f,g\colon X\rightarrow Y$ are homotopic if and only if $f=g$.

$X$ is a connected space and $Y$ is a discrete space prove that the two maps $f,g\colon X\rightarrow Y$ are homotopic if and only if $f=g$. I am trying to solve few problems in algebraic topology, but I don't have deep knowledge in the subject. I…
junegirl
  • 61
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Question on prop 0.18 by Hatcher algebraic topology

Here is prop 0.18 from Hatcher: if $(X_1,A)$ is CW pair and we have attaching maps $f,g:A\rightarrow X_0$ that are homotopic, then $X_0\sqcup_f X_1$ is homotopy equivalent to $X_0\sqcup_g X_1$ relative to $X_0$. Hatcher says if $F:A\times…
onto heart
  • 69
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If f~ g then $ \pi_0(f) =\pi_0(g)$

This question was left as an exercise in my class of Algebraic Topology and I am struck on on of it parts. So, I am posting it here. Question: For all topological spaces X, we let $\pi_0(X) $ denote the set of arcwise connected components of X. …
user775699
3
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1 answer

Group Action of $\mathbb{Z}$ on $\mathbb{R}^2\setminus \{0\}$ and covering space

if the group $G=\mathbb{Z}$ acts on $Y=\mathbb{R}^2\setminus \{0\}$ using by the action $m\cdot (x,y)=(2^m x, 2^m y)$ then I was able to show that $Y \to Y/ \mathbb{Z}$ is a covering map. Can I conclude that $\pi_1(Y/\mathbb{Z})=G=\mathbb{Z}?$ I…
permutation_matrix
  • 1,356
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Some Quotient Space $(\prod_{i=1}^n S^2/\bigvee_{i=1}^n S^2 )^4= \bigvee_{1\leq i< j\leq n } S^4$

I want to know a 4-skelecton of quotient space $(\prod_{i=1}^n S^2/\bigvee_{i=1}^n S^2 )^4 = \bigvee_{1\leq i < j\leq n} S^4$ If $n=2$, then I can realize. But in $ n=3$ case I have no idea. Give me a hint. Thank you in advance. [$n$=2] $S^2\times…
HK Lee
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