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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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Hatcher 1.2.4 solution?

Problem 1.2.4: Let $X \subset \mathbb{R}^3$ be the union of $n$ lines through the origin. Compute $\pi_1 (\mathbb{R}^3 - X)$. I have the following solution. Would someone be so kind as to check whether it's correct? Also, I have seen that there are…
alcithoe
  • 145
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Deciding whether or not Two Circles are Linked

Consider the two-dimensional vector subspaces $V_1, V_2 \subseteq \mathbb{R}^4$ given by \begin{align} V_1 & = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_3 = x_4 = 0 \} \\ V_2 & = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_2 = 0 \}.…
Alan Yan
  • 1,456
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Excisive triads and weak equivalences

Let us suppose that $f(X',A',B',x_0') \rightarrow (X;A,B,x_0)$ is a map of triads such that $$f_\ast:\pi_\ast (A' \cap B',x_0') \rightarrow \pi_\ast(A \cap B, x_0)$$ $$f_\ast:\pi_\ast(A',x_0')\rightarrow \pi_\ast(A,x_0)$$ and $$f_\ast :…
Tedar
  • 529
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Serre Spectral Sequence with Local Coefficients: An Example

My goal is to give a concrete calculation with the Serre spectral sequence where the coefficient system is not simple. Here's what I have so far: Take X = $S^1 \times S^2 / \mathbb Z_2$ where $\mathbb Z_2$ acts as the antipodal map on both factors.…
Mathis
  • 205
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Not fully understanding the definition of topological invariance

Let me start by saying that I'm a student self-studying this for fun so I might be asking a really silly question here, anyways: According to Wikipedia, a topological invariant is defined as a property of a topological space that remains invariant…
SourBiscuit
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Homology of the Euclidean space with a closed set removed

Suppose that $n\in\mathbb{Z}$ and $n\geq 1$. Let $A$ be a closed subset of $\mathbb{R}^n$. Show that for all $k\in\mathbb{Z}$ and $k\geq 0$, we have $\widetilde{H}_{k+1}(\mathbb{R}^{n+1}-A)\simeq\widetilde{H}_{k}(\mathbb{R}^{n}-A)$, where…
Jianxing
  • 141
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Fundamental group of a mapping torus

I've read through similar questions and have some additional questions. Exercise: Let $f:X\rightarrow X$ be a self-mapping. Let $X$ be simply connected. Define mapping torus $T_f$ as pushout of $$\require{AMScd} \begin{CD} X\times \{0,1\} @>{i}>> X…
average math enjoyer
  • 150
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Why is cellular homology considered more efficient computationally?

Suppose that we want to find out cellular homology groups of $RP^n=:X$. I know that $RP^n$ is obtained by attaching $k$ cell to $RP^{k-1}$ for every $k$ such that $0\le k\le n$. So the chain complexes are $C_j(X)=H_j(X^j, X^{j-1})=Z$ for every $j$…
Koro
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Non trivial 1-cocycle on the circle

I read Example 9.1. Of "Differential forms in Algebraic Topology" Bott&Tu's. We consider the following Good cover of the circle And we study they Čech cohomology which is the cohomology induced by the coboundary operator…
Joniloli
  • 111
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Understanding Rotman Algebraic Topology Theorem 10.46

Definitions and background result Theorem Question What exactly is the isomorphism ("via" $p_*$) with $G \cong \pi_1 (\tilde{X}_G, \tilde{x}_0)$? It cannot be $p_*$ restricted to its image, i.e. $\tilde{p}_* : \pi_1 (\tilde{X}_G, \tilde{x}_0)…
IsaacR24
  • 635
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Show a contractible compact subset of $\Bbb{R}^2$ has connected complement.

Let $A \subset \Bbb{R}^2$ be compact and contractible. I need to show $\Bbb{R}^2 \setminus A$ is connected. I know since $A$ is a compact subset of the plane, it is closed and bounded thus its complement is open. Do I also need to show its closed to…
homosapien
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Clarifying's Rotman's proof of $\pi(K, p) \cong G_{K,T}$

I understand most of this proof, but there's some definitions that are not completely clear to me. I'll provide only the relevant snippet of the proof the confuses me. Above the proof, I provide relevant definitions. 1. Definitions and Proof 2.…
IsaacR24
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Questions about orientability of a sphere bundle

I am having some trouble with Bott–Tu's definition of orientability of a sphere bundle. I've pasted the definition in as a picture below. In particular, in the first sentence of the second paragraph, they say that "a generator of the top cohomology…
boink
  • 1,438
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The intuitive proof of $[\Sigma X, K] = [X,\Omega K]$

I want to know the proof of $[\Sigma X, K] = [X,\Omega K]$ which is called "adjoint relation " Here $\Sigma X = SX/\{x_0\}\times I$, and $\Omega K$ is a space of loops in $K$ at chosen base point. And $[Y,Z]$ is a set of homotopy classes of a map…
HK Lee
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$H_i(X \times S^n) \cong H_i(X) \oplus H_{i-n}(X)$ for all $i,n$?

How should I prove that $H_i(X \times S^n) \cong H_i(X) \oplus H_{i-n}(X)$ for all $i,n$? What was my approach. I am not allowed to use Kunneth's Formula because it would be too easy. Rather I think I can use Mayer-Vietoris Sequence. Consider the…
permutation_matrix
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