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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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Is it ever true that $\Omega^n(\bigvee_I X)\simeq\bigvee_I\Omega^n X$?

I'm curious about the following: Given a countable index set $I$, is it ever true that $\Omega^n(\bigvee_I X)\simeq\bigvee_I\Omega^n X$? What would we have to assume about $X$ (if possible) to make it true? My motivation (wishful thinking) is that…
Eivind Dahl
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Use of Non-Reduced Homology

This is the dual of Use of Reduced Homology. The answers to that question and the fact that basically every homological argument I have seen so far gets easier when using reduced homology left me with the question: Why do we introduce non-reduced…
Eike Schulte
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Induced map by $\mathbb{R}P^2\to\mathbb{C}P^2$ in homology.

Let $i\colon \mathbb{R}P^2\to \mathbb{C}P^2$ the usual embedding $i( [x:y:z] )=[x:y:z]$. I want to compute the induced map by $i$ in the second homology groups with $\mathbb{Z}/2$ coefficients. I think the easiest way should be to use the…
Laszlo
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Showing that an inclusion is null homotopic

I'm trying to do exercise 5 on page 18 in Hatcher: Show that if a space $X$ deformation retracts to a point $x \in X$, then for each neighborhood $U$ of $x$ in $X$ there exists a neighborhood $V \subset U$ of $x$ such that the inclusion map $V…
Rudy the Reindeer
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A question from cellular homology.

Let X be the 2 complex obtained from $S^{1}$ with its usual cell structure by attaching two 2 cells by maps of degrees 2 and 3 , respectively. (a) Compute the homology groups of all the subcomplexes $A ⊂ X $ and the corre- sponding quotient…
user51266
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Computing the fundamental group of the suspension of $X=\{0,1,1/2,1/3,1/4,...\}$.

This is the problem 1.2.18 from Hatcher's algebraic topology. Let $X=\{0,1,1/2,1/3,1/4,...\}$, inheriting a subspace topology from the real line, and $SX$ be its suspension, which looks like this: We need to show that its fundamental group is free…
JSCB
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What is meant by 'passage from "continuous mathematics" to "discrete mathematics"'?

In Tammo tom Dieck's 'Algebraic Topology' (corrected, 2nd printing from 2010) on page 24, it says The passage from TOP to h-TOP may be interpreted as a passage from "continuous mathematics" to "discrete mathematics". and I have discovered…
polynomial_donut
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concerning the definition of homotopy extension property

1) Let $X$ be a topological space, and let $A \subset X$. We say that the pair $(X,A)$ has the homotopy extension property if, given a homotopy $f_t\colon A \rightarrow Y$ and a map $\tilde{f}_0\colon X \rightarrow Y$ such that $\tilde{f}_0 |_A =…
Samir
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Is it possible that isomorphic $\pi_n$'s not induced by a map?

This might be a stupid question, but I want to make sure as a beginner of AT. A map between CW-complexes $f: A \rightarrow B$ is defined to be a weak homotopy equivalence if it induces isomorphisms $f_*: \pi_n(A) \rightarrow \pi_n(B)$ for all $n$.…
Clueless Gorilla
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Simply connected reduced suspension on path connected X

If X is path connected how may i show that the reduced Suspension $\Sigma $ X is then simply connected. I cannot seem to picture this construction
Desmodr
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Retraction of a closed orientable surface onto a simple closed curve

Let $S$ be a closed, orientable surface and $\gamma$ a simple closed curve on $S$. Show that $S$ retracts onto $\gamma$ if and only if $\gamma$ represents a non-trivial element in homology. Clearly, if $[\gamma]=0$ in $H_1(S)$, the…
abrax
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Universal Cover of $\Bbb{R}\textrm{P}^2 \vee \Bbb{R}\textrm{P}^2$ in Hatcher

On page 78 of Hatcher here he constructs an example of the universal covering space of $\Bbb{R}P^2 \vee \Bbb{R}P^2$. Now from the picture in example 1.48 I think I understand why it is a covering space. Our covering space map $p$ is the usual…
user38268
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Surface of genus g is not homotopy equivalent to wedge of cell complexes both with non-trival H_1

I came across this question while studying for a qualifying exam: Prove that a closed orientable surface of genus $g \ge 1$ is not homotopy equivalent to the wedge $X \vee Y$ of two finite cell complexes both of which have nontrivial…
paragon
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Can $\mathbb{Z}/6\mathbb{Z}$ act freely and properly discontinuously on $\Sigma_4$?

Let $\Sigma_m$ denote the closed connected orientable surface of genus $m$. Let $N_m$ denote the closed connected non-orientable surface of genus $m$. I was wondering which cyclic groups could act freely and properly discontinuously on $\Sigma_4$.…
Mr. Frog
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Hatcher problem 1.2.3 - technicality in proof of simply connectedness

I am trying to prove that $\Bbb{R}^n$ minus finitely many points $x_1,\ldots,x_m$ is simply connected, where $n \geq 3$. For days now I have tried many different arguments but I have found flaws in all of them. I have finally come up with one,…
user38268
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