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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.

Please use more specific tags like , ,, or whenever appropriate.

21356 questions
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Non-normal covering of a Klein bottle by torus.

I am trying to construct a non-normal covering of Klein bottle with itself and by a torus. For Klein bottle to Klein bottle, I got a three sheeted covering, just glue three Klein bottles, which is non-normal but whatever covering I am constructing…
ovi
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Zeroth homology and cohomology of the rationals

I'm asked to show that in singular homology and cohomology $H_0(\mathbb{Q},\mathbb{Z}) \ncong H^0(\mathbb{Q},\mathbb{Z})$, $\mathbb{Q}$ being endowed with the subspace topology from $\mathbb{R}$. Since $\mathbb{Q}$ is totally disconnected, the…
Kevin Carlson
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Homology groups of a tetrahedron

I have been solving a lot of questions lately in preparation for an exam which is in about 3weeks. Well, I came across this question but I have no clue. I think I would need a lot of help. I am asked to compute the homology groups of a tetrahedron…
smanoos
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Hatcher Problem 1.2.11: Cell decomposition of Mapping Torus $T_f$ of $S^1 \times S^1$

Suppose we have continuous function $f : X \to X$ that sends the basepoint of $X$ to itself, viz. $f(x_0) = x_0$ where $x_0$ is the basepoint of $X$. Recall the definition of the mapping torus $T_f$ of a space $X$. It is defined to be $X \times I$…
user38268
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Finding the fundamental group of the complement of a certain graph

I'm having trouble solving problem 12 from Section 1.2 in Hatcher's "Algebraic Topology". Here's the relevant image for the problem: I'm trying to find $\pi_1(R^3-Z)$, where $Z$ is the graph shown in the first figure. The answer (according to the…
alephzero314
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Torsion in homology groups of a topological space

It seems as though "nice" spaces don't have torsion in their homology groups. What is the underlying characteristic of these nice spaces; that they can be embedded in $\mathbb{R}^3$? So what are some examples of spaces which can be embedded in…
user33212
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Homotopy classes of functions from a finite CW complex

I am given the following problem: taken $X$ finite CW complex and $Y$ a space such that for every basepoint $y \in Y$ the group $\pi_i(Y,y)$ is finite $\forall i \leq \text{dim} X$ then the set $[X, Y]$ is finite. My first approach was the…
N.B.
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Mayer-Vietoris implies Excision

Assume $H_n$ is a covariant homotopy functor on the category of locally compact Hausdorff spaces which has the Mayer-Vietoris property: whenever $X$ is the union of two closed subspaces $A$ and $B$ there is a long exact sequence $$\to H_n(A \cap B)…
Paul Siegel
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Does the ham sandwich theorem hold for dividing objects into thirds?

The ham sandwich theorem states that given $n$ measurable "objects" in $n$-dimensional space, it is possible to divide all of them in half (with respect to their measure) with a single $(n−1)$-dimensional hyperplane. In $n$-dimensional…
Patrick Shambayati
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Homotopy type of a space

I was wondering what tools of algebraic topology are usually used to show that some things have the same homotopy type? Hatcher doesn't really talk about this in his book even though he defines the concept on page 3. Of course we can compute the…
asdf
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Intuition for homology of $S^n$

We know that the sphere $S^n$ has $n$-th singular homology $H_n(S^n)= \mathbb{Z}$. A generator is given by a fundamental class, which is nothing else than the sum of the simplices in some triangulation. Thus my question is: Is there any intuition…
phil
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Showing that $\pi(M \# N) = \pi(M) \ast \pi(N)$ for $n$-dimensional manifolds $M$,$N$

Problem: Let $M$ and $N$ be $n$-dimensional manifolds, where $n > 2$. Let $M \# N$ be their connected sum. Show that $\pi(M \# N) = \pi(M) \ast \pi(N)$. RE-EDITED Attempt: Let $U_2$ and $V_2$ be two small open balls from $M$ and $N$ to be…
user1770201
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Topology of a cone of $\mathbb R\mathbb P^2$.

I had already posted this on mathoverflow and was advised to post the same here. So here it goes: $X=\{(x,y,z)|x^2+y^2+z^2\le 1$ and $z≥0\}$ i.e. $X$ is the top half of a $3$-Disk. $Z=X/E$, where $E$ is the equivalence relation on the the plane $z =…
Will
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Homology with local coefficients in a $\mathbb{Z}[\pi_1(X)]$-module

I am confused about this "homology with local coefficients" business. First, I am confused by its name. What is "local" about it? Second: Let $X$ be a topological space with a universal cover. Let $G=\pi_1(X)$. Let $M$ be a $G$-module, that is, a…
Bruno Stonek
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A Question about Borsuk-Ulam Theorem

I don't understand a step of Borsuk-Ulam theorem, which i tagged with a star below. $\underline{Borsuk-Ulam}$: If $f:S^2\rightarrow\mathbb R^2$ continuous, then $\exists x$, s.t. $f(x)=f(-x)$ according to the proof(by contradiction): Assume, there…
derivative
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