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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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The cofibre of a type $n$ $p$-local CW complex has type $n+1$

I am trying to prove this relatively simple statement from 'Nilpotence and periodicity in Stable Homotopy Theory' Suppose $X$ as in the periodicity theorem has type $n$. Then the cofibre of the map 1.54 has type $n+1$ The definitions needed: $X$…
Juan S
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Universal Coefficient Theorem for Coefficient Group ${\bf R}$

${\rm Ext}\ ({\bf Z}_n,G) =G/nG$ so that if $G={\bf Q},\ {\bf R}$ then ${\rm Ext}\ ({\bf Z}_n,G)=0$ Hence UCT implies $$ H^n(C;G) = {\rm Hom}_{\bf Z} ( H_n(C);G) $$ Hence $$ H^n(C;G) =H_n(C)/{\rm Tor}(H_n(C)) \ \otimes_{\bf Z} G$$ Why we do consider…
HK Lee
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f~g iff if $f*\bar{g}$~$\varepsilon_x$

Can you please help me with this question? let f,g:I $\rightarrow$ X be two paths in X from x to y. Prove that f~g iff $f*\bar{g}$~$\varepsilon_x$ (where $\bar{g}$=g(t-1)) Thanks in advance
user135541
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strong deformation retract, of a perforated plane?

Let $x_0 \in R^2$. How do I find a circle in $R^2$ which is a strong deformation retract of $R^2-\{x_0\}$? Is it just a unit circle with the center $x_0$?
user135541
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Path components of an $H$-space

From Hatcher, page 291, page 3. Show that if $X$ is an $H$-space such that the set of path-components of $X$ is a group with respect to the multiplication induced by the $H$-space structure, then all path components are homotopy equivalent. I am…
Bombyx mori
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Continuous mapping from n-sphere to (n+1)-sphere

Are there any "nice" functions that can take a point from the surface of an n-sphere and map it to a the surface of an (n+1)-sphere? By "nice", I mean it should be continuous, one-to-one (but not necessarily onto), and cover lots of surface area…
Aaron Voelker
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Showing that $\left\langle a,b \mid abab^{-1}\right\rangle \cong \pi_1(K) \cong \left\langle c,d \mid c^2 d^2 \right\rangle$

Hypothesis: Let $G$ and $H$ be defined in terms of the following presentations: $$ G \cong \left\langle a,b \mid abab^{-1}\right\rangle $$ $$ H \cong \left\langle c,d \mid c^2 d^2 \right\rangle $$ Goal: Show that $G \cong H$. Hint: Show that $G…
user125103
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how does dunce cap has simplicial structure

Dunce cap is a example of space which is contractible but not collapsible, but collapsibility is only defined for simplicial complexes. Can anyone explain how does dunce hat has simplicial structure?
user93620
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Basic property of homotopy

Suppose $ f : X \to Y$ and $g,h : Y \to Z$ are continuous, with $g \simeq h$. Prove that $ gf \simeq hf $. My attempt: Suppose $ L(x,t) $ gives a homotopy from $g$ to $h$, i.e. $ L(x,t) : Y \times I \to Z $, with $ L(x,0) = g(x)$ and $ L(x,1) =…
TRY
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Examples of fundamental group of mapping torus

The original question was askes here. I donot know how to apply or compute any example. I think a specified explanation will be helpful. Let $M=\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and $f=A\in\mathrm{SL}(2,\mathbb{Z})$ be the quotient action on…
Pengfei
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Mapping Cones in the Stable Homotopy Category

This question comes from the text Stable Homotopy and Generalised Homology by Adams and so the "stable homotopy category" here refers to his construction via CW-spectra. With this framework in mind, at one point Adams defines the mapping cone of an…
James Miller
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Why cellular maps induce maps of chain complexes?

If $X$ is a CW-complex and $f:X\rightarrow X$ a cellular map. Then why it induces a map of chain complexes $f_*:C_*(X)\rightarrow C_*(X)$. (Why it commutes with the differential?).
Berry
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CW complexes exercise from Hatcher

I am having some trouble with this. Any help would be very appreciated. Thanks. Exercise 24. Let $X$ and $Y$ be $CW$ complexes with $0$ cells $x_{0}$ and $y_{0}$. Show that the quotient spaces $X * Y / (X*\left\{y_{0}\right\} \cup…
Anna
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fundamental groups of open subsets of the plane

This should be a very basic algebraic topology question. The other day I was thinking about the fact that $P^2(R)$ has $\pi_1 = Z/2Z$. On the other hand I thought to myself how something like this can never happen for, say, an open subset of the…
user125763
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What is a loop in $\mathbb RP^n$

What is a loop in $\mathbb RP^n$ ? I have to show that: Given a loop $\alpha:[0,1]\rightarrow\mathbb RP^n$ starting and ending in $x_0=[N]=[S]$ and its lift $\tilde{\alpha}:[0,1]\rightarrow S^n$ starting from $N$ Prove that; $\chi:\pi_1(\mathbb…
derivative
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