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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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Using retraction for show that:

Let $f:\mathbb{S^2} \rightarrow \mathbb{R^2} \diagdown \{(0,0)\}$ a continuous application. Proof that there is $(x_0,y_0,z_0)\in \mathbb{S^2}$ such that $f(x_0,y_0,z_0)=\lambda(x_0,y_0)$ for some $\lambda \in \mathbb{R} $.
Henfe
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Retraction Problem

Here is a qual problem that I am really struggling with. The only method I know is the standard fundamental group trick such as in how one shows that there is no retraction from the disk to the circle. Any help is appreciated If $A$ is a subspace…
Kristie
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Compute the homology of the CW complex directly from the cell structure

Let $X$ be a CW-complex. Define an equivalence class on $X$ to be $\alpha(X)$, and $Y \in \alpha(X) \iff Y$ homotopy equivalent to $X$. Define an ordering on cell structures: The cell structure with a smaller number of cells is smaller. E.g. $e_0…
Dávid Natingga
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Hairy Ball Theorem and $SO(3)$

How to prove that $SO(3) \neq S^2 \times S^1$ using Hairy Ball Theorem? In other words, if assuming $SO(3) = S^2 \times S^1$, how to construct a non-vanishing vector field on $S^2$?
hb12ah
  • 399
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On the homotopy type of unions of 2-spheres

Here is the problem I am stuck on: If $X$ is a connected Hausdorff space that is a union of a finite number of $2$-spheres, any two of which intersect in at most one point, then show that $X$ is homotopy equivalent to a wedge sum of $S^1$ and…
James Miller
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$g_\ast : H_1(X; {\bf Z}) \rightarrow H_1(X;{\bf Z})$

The following is the exercise in Algebraic topology course. Assume that $ X$ is a finite connected graph such that no vertex is the endpoint of just one edge. Let $G$ be a group acting $X$. Then for any $g\in G$, we have $$g_\ast : H_1(X; {\bf Z})…
HK Lee
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An oriented Grassmannian is a product of two spheres

How to prove that the Grassmannian of oriented subspaces $\mathrm{Gr}_+(2,4,\mathbb R)$ is homeomorphic to $S^2\times S^2$? I know that $\mathrm{Gr}_+(2,4,\mathbb R)\cong\mathrm{SO}(4)/(\mathrm{SO}(2)\times\mathrm{SO}(2))$, but don't know whether…
danneks
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$H_1$ of solid torus is $\mathbb Z$ or $\mathbb Z^2$?

I have what seems like a very silly argument, but I can't figure out what's wrong with it. Suppose that I'm trying to calculate $H_1$ of a solid torus $X$ (a torus with its ``interior'' filled in). On the one hand, it's homotopy equivalent to a…
fish
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Connecting morphism in the long exact sequence of homotopy groups for a fibration

I'm reading Bott and Tu's book "Differential forms in Algebraic Topology" and I need some help to understand a detail on the long exact sequence (LES) of homotopy groups for a (Hurewicz or Serre) fibration. Given a base point preserving fibration…
Mathex
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Hatcher 2.1 problem 19 - homology of square boundary plus interior points with first coordinate rational

I need help verifying and completing my solution to problem 2.1.19 of Hatcher's book Algebraic Topology. Calculate the homology groups of the subspace of $I \times I$ consisting of its 4 boundary edges and all the points in its interior with…
abhi01nat
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universal coefficient theorem for cohomology

We all know that we can compute homology and cohomology with arbitrary coefficient if we already know the homology groups with coefficient in $\mathbb{Z}$. I wonder if it is possible if we know the cohomology groups with coefficient in some groups,…
Qijun Tan
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The Relationship Between Topological Space and Fundamental Group

I am just an Algebraic Topology beginner and I have one doubt related to the relationship between the topological space and the fundamental group. As we know that if given a specific topological space, we can calculate the fundamental group of it.…
David James
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Chain complex and its homomorphisms

The definition of a chain complex in algebraic topology requires that $d_n\circ d_{n+1}=0$, but why is that? It feels somewhat arbitrary, I have not found it motivated, and it is not obvious to me that this is an interesting constraint. Why not…
Frank
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Computing the degree of specific maps on spheres

I am trying to understand lens spaces. While understanding the basic definition is rather easy, everything that follows appears to be quite hard. I wanted to understand the classification up to homotopy equivalence, which is outlined as two…
Sellerie
  • 553
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Let $X = S^1 \times S^1$ and let $p_1,p_2,p_3 \in S^1$ and let $A =....$. Compute $H_i(X,A)$ $\forall i$.

Let $X = S^1 \times S^1$ and let $p_1,p_2,p_3$ be distinct points in $S^1$ and let $A = (S^1 \times \{p_1\}) \cup (S^1 \times \{p_2\}) \cup (S^1 \times \{p_3\})$ Compute $H_i(X,A)$ $\forall i$. We have a L.E.S.: $$0 \rightarrow H_2(A)…
user637978
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