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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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Two questions on Massey's Algebraic Topology, Chapter 2.8

The first qusetion is about the following proposition. Let $E_1$ and $E_2$ be closed discs with boundaries $B_1$ and $B_2$, respectively. Then, any continuous map $\mathit f$ : $B_1$$\to$$B_2$ can be extended to a contiuous map F : $E_1$$\to$$E_2$.…
GTM 73
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Defining cellular homology independent of singular homology

Can cellular homology be defined without using simplicial or singular homology? One obstruction is to compute degree of attaching maps without invoking any homology theory.
Arun
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Fundamental group of the wedge sum of two spaces

Let $X,Y$ be two path-connected topological spaces and $\langle A\mid R\rangle,\langle B\mid S\rangle$ respectively presentations for their fundamental groups. I think that a presentation for the fundamental group of the wedge sum $X\vee_{x_{0}} Y$…
Her
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Is a torus homeomorphic to a cylinder?

For a cylinder as the way $S^1 \times [0,1]$, both surfaces are orientable and with Euler characteristic 0. So they are homeomorphic, yeah? But they have different fundamental group. So they are not homeomorphic?? Please, help me. Thanks!
LH8
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What is the motivation for the "Covering Homotopy Property" in a fibration?

Basically a beginner type of topology question here, but I am trying to understand something and am a bit stuck on a definition. According to J.P. May, a fibration is a map $p : E \to B$ such that for all spaces $Y$, that embed into $E$ by some map…
Mikola
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Homotopic maps induce the same homomorphism

This exercise is from Kosniowski's Algebraic Topology, page 137, exercise 15.11, d. Show that two continuous maps $\varphi, \psi :X \rightarrow Y$ with $\varphi(x_0) = \psi (x_0)$ for some $x_0 \in X$ induce the same homomorphism from $\pi(X,x_0)$…
Yagger
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Calculating homology of mapping torus (is this question incorrect?)

This question is on an old qualifying exam for my institution: Let $Y$ be a space and $f:Y\to Y$ a self-mapping. Let $X$ be the mapping torus of $f$ (i.e. the space obtained from $Y\times I$ by identifying $(y,1)$ and $(f(y),0)$ for $y\in Y$).…
Alex Mathers
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Calculating H_0 directly from Eilenberg-Steenrod axioms

It's well-known that every homology theory satisfying Eilenberg-Steenrod axioms is isomorphic to singular homology. I tried to perform some homology calculations directly from axioms but couldn't do even a simple task: I can't prove that $H_0$ of a…
Dmitry
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help-need to determine that this induced map is the zero map

Let $f : S^3 \rightarrow S^3$ have the property $f(x) = f(-x)$ for every $x \in S^3$. Show that $f_{*} : H_{3}S^3 \rightarrow H_{3}S^3$ is the zero map.
alfredo smith
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Computing $H_*((S^n\times S^m)/\mathbb{Z}_2)$

What is the fastest way to compute $$H_*((S^n\times S^m)/\mathbb{Z}_2;\mathbb{Z}),$$ where $\mathbb{Z}_2$ acts on both factors by the antipode map? Is there a better way than using the Serre spectral sequence?
Sophie
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$\pi_1(\mathbb{R}^n -A) \cong \pi_1(\mathbb{R}^{n+1} -A) \Rightarrow \pi_1(\mathbb{R}^n -A) \cong \pi_1(\mathbb{R}^{n+k} -A) \ (k \geq 1)$?

Let $A$ be a subspace of $\mathbb{R}^{n-1}$ with $n \geq 4$. And if we know $\pi_1(\mathbb{R}^n -A) \cong \pi_1(\mathbb{R}^{n+1} -A)$, then can we say that $\pi_1(\mathbb{R}^n -A) \cong \pi_1(\mathbb{R}^{n+k} -A)$ for all $k \geq 1, k \in…
Junyu
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$X,Y$ are homotopy equivalent, so the number of connected component in $X$ and $Y$ is equal

$(X,\mathcal{T}),(Y,\mathcal{O})$ are homotopy equivalent, denote the homotopy equivalent functions by $f$ and $g$ ($f\circ g\simeq Id_Y, g\circ f\simeq Id_X$). from $f,g$ continuity , taking a connected component $[x]$ (A maximal connected subset…
user5721565
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Homotopy fiber of inclusion of projective spaces equivalent to sphere $S^3$

Consider the inclusion map $S^2=\mathbb{C}P^1 \overset{f}{\to} \mathbb{C}P^\infty$ ($\mathbb{C}P^\infty$ is the sum [direct limit] of $\mathbb{C}P^n$s) and the mapping space $E_f\subseteq \mathbb{C}P^1 \times {(\mathbb{C}P^\infty)}^I$ with a…
savick01
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Rotman Introduction to Algebraic Topology Question 0.5

I've been stuck on a problem from Rotman's Introduction to Algebraic Topology for a while. I'm doing the exercises outside of class right now so it's difficult to ask for help. I'm hoping someone here can help out. The question is: Let $f,g:I \to I…
Mmhmm
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A surface of genus g is not homotopy equivalent to a wedge sum of CW-complexes

I found some trouble solving an exercise from Hatcher (n. 18, p. 230). It asks to show that a closed, orientable surface $M$ of genus $g$ is not homotopy equivalent to a wedge sum of CW-complexes $X$ and $Y$ with non-trivial reduced homology. I…
Lukath
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