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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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Showing that a CW space is contractible if it is endowed with a certain binary operation

I am having trouble with the following homework problem, and was hoping someone could provide me with a hint: I am given a connected CW space $X$ which has a continuous associative operation $(x,\ y)\mapsto x\circ y$. It is also given that $x\circ…
Roy Ben-Abraham
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Possible cup product structures on a manifold

I am studying for a qualifying exam, and I came across this problem: Let $M$ be a closed orientable connected 4-manifold with $H^1(M) = H^3(M) = 0$ and $H^2(M) \cong H^4(M) \cong \mathbb Z$. What are the possible cup product structures on…
paragon
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Fundamental group of the torus

I'm trying to get my head round the following calculation of the fundamental group of the torus, using Seifert Van-Kampen (I know it's easier to do this by considering covering spaces, but I'm trying to learn the Seifert Van-Kampen method).…
Martin
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Difference between homotopy equivalence and based homotopy equivalence?

As the question suggests, what is the difference between homotopy equivalence and based homotopy equivalence?
user236251
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Existence of a simple closed curve which is not null-homotopic

Problem. Assume that $U$ is an open and connected subset of $\mathbb R^2$, and $\gamma :[0,1]\to U$ is a closed curve, which is not null-homotopic in $U$ and not necessarily simple closed. Show that there exists a simple closed curve, which is not…
Yiorgos S. Smyrlis
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Hopf invariant formula

I'm having some trouble proving the following statement: let $g:S^{2n-1} \to S^{2n-1}, f: S^{2n-1} \to S^n$. Then $H(f\circ g) = \deg g H(f)$ where $H(f)$ is the Hopf invariant. The definition I am using for Hopf invariant is as follows: let $C_f…
Eric O. Korman
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Non-realizability of $\mathbb{Q}$ as a cohomology group

In the paper "On the realizability of singular cohomology groups" by Kan and Whitehead, it is shown that there is no space $X$ and integer $n\geq 1$ such that $H^{n-1}(X)=0$ and $H^n(X)=\mathbb{Q}$ (cohomology with integral coefficients). At the…
Aldo Guzmán Sáenz
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No retract $X \wedge \mathbb{R}P^2 \to X \wedge \mathbb{R}P^1$

Let $X$ be a finite CW complex, and suppose $\Sigma X \cong X \wedge \mathbb{R}P^1$ is not contractible. By considering the fundamental group or otherwise, it is easy to see that there can be no retraction $\mathbb{R}P^2 \to \mathbb{R}P^1$. But…
JHF
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Fundamental group of a quotient on a solid torus.

It is easy to compute the fundamental group of a solid torus. You easily get $\mathbb{Z}$ just because the torus is the cartesian product of a circumference and a closed disk. The next step is imagine two solid torus one inside of the other one. Now…
DCao
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Poincare duality isomorphism problem in the book "characteristic classes"

This is the problem from the book, "characteristic classes" written by J.W. Milnor. [Problem 11-C] Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow A$. Let $k = p-n$. Show that the Poincare…
ljh8372
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Continuously deform 2-torus with a line through one hole to make it go through both

The problem is in this video at 18:00. If you have a 2-holed 2D torus with a line ($\mathbb{A}^1$) going through one of its holes. How do you deform it to make the line go through both holes.
aelguindy
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Why is $\pi_* MU$ concentrated in even degrees?

$\pi_* MU$, which is the cobordism ring of manifolds with a complex structure on the stable normal bundle, is a polynomial ring $\mathbb{Z}[x_2, x_4, \dots]$. I'm probably being silly here, but is there some obvious reason why everything is in even…
Akhil Mathew
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Homeomorphic, homotopy equivalent and deformation retracts. How do I get a feeling for this?

We have homeomorphism, homotopy equivalences and deformation retracts ( which are a particular case of the latter). Now my problem is that I know what they all mean, but I have troubles to see them in real world objects. Imagining somebody would…
user159356
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Motivation behind definition of homologous cycles

Two cycles are said to be homologous if their difference is a boundary (usual meanings implied). What is the motivation behind this definition or the intuitive meaning it carries? I am looking of something along the line of definition of homotopic…
Isomorphic
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Question from Munkres algebraic topology section 58: retractions

This is question 7 on page 366 from section 58 of Munkres Topology: Let $A$ be a subspace of $X$, let $j: A \to X$ the inclusion map, $f:X \to A$ continuous. Suppose there is a homotopy $H$ between $j \circ f$ and the identity on $X$. a) Show if…
Johnny Apple
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