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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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there is no 3-manifold such its fundamental group is isomorphic to direct product Z⊕Z⊕Z⊕Z.

Prove that there is no 3-manifold such its fundamental group is isomorphic to Z⊕Z⊕Z⊕Z.I tried to use Betti number or use the theorem that says if it is 3-manifold then if it is orientable any Alexander module produced in G is self-reciprocal and if…
user466441
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Covering spaces of $X=(S^1\times S^1)\vee S^1$?

Let $X=(S^1\times S^1)\vee S^1$. What are all connected 3-sheeted covering spaces of $X$ up to homeomorphism. I found one which is three squares with three circles from the universal covering. But I am not sure it is right. My guess about the…
Ling
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derivation of cohomology of a torus

Let $V$ be an $n$-dimensional complex vector space and $U \subseteq V$ a full dimensional lattice (i.e. $U \cong \mathbb{Z}^{2n}$) and let $X=V/U$. Something I'm reading says "since $V$ is contractible, $H^1(X, \mathbb{Z}) = Hom(U, \mathbb{Z})$".…
usr0192
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Degree of map from $S^n$ to $S^n$

This is an excercise in the book Algebraic Topology - Greenberg and Harper Excercise : Let $f$ and $g$ be a map from $S^n$ to $S^n$ such that $f(x)\neq g(x)$ for all $x$ Then $f $ is homotopic to $ag$ where $a$ is antipodal map, hence…
HK Lee
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Computation of Cellular Homology

I have a few questions about computing cellular homology. Setup: Here is an outline of the process of computing cellular homology as far as I understand it: Put a CW complex structure on your space $X$. Form the cellular chain complex $$\ldots…
Chris
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Euler and Betti numbers' relation

Is there an intuitive reasoning behind the relation of topological Euler characteristic and Betti numbers - the former being an alternating sum of the latter (starting with a plus sign at that)? Or is it purely coincidental?
Kosm
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Cup Product - Leibniz Rule

I'm trying to prove the Leibniz Rule: $$\partial(u \smile v) = \partial(u) \smile v + (-1)^{|u|} \ u \smile \partial(v) $$ and I'm having some difficulties. Here's my attempt: Let $|u| = p$, $|v|=q$ and $\sigma: \Delta^{p+q+1} \longrightarrow…
user362073
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Line with two origins and higher homotopy groups

So I know how the universal cover of the line with two origins is(looks like a comb going in both directions with alternating midline points corresponding to the different origins). I was wondering why the higher homotopy groups vanish in the…
Enigma
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Covering space of a wedge of two circles

Next question: Let $X$ be a wedge of two circles, $X=A \vee B$ and let $x_0$ denote the point of tangency. Let $q:S^1 \to X$ be the identification map that identifies -1 and 1 $\left(q\left(-1\right)=x_0=q\left(1\right)\right)$ The question is to…
Juan S
  • 10,268
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Simple example that $H_{1}(\mathcal{X})=0$ but $\mathcal{X}$ is still not orientable.

Can anyone show me a simple example of a manifold such that $H_{1}(\mathcal{X})=0$ but $\mathcal{X}$ is not orientable? We know the contention hold for $\pi_{1}$, I am not sure if it holds for $H_{1}$.
Bombyx mori
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$\pi_1({G,1})$ is abelian if $G$ is a topological group: understanding a proof

I am reading a proof about the fact that the fundamental group $\pi_1({G,1})$ is abelian if $G$ is a topological group. The proof starts by taking two loops $f,g$ at $1$ to define the function $H:I^2\to G, H(s,t)=f(s)g(t).$ The autor say that "$H$…
Alex
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Cogroup objects in topological spaces

Let A be a pointed topological space. I want to show that A is a cogroup object in $Ho(Top_*)$ iff the functor $[A,\_] \colon Top_* \rightarrow Sets_*$ factors through the category of groups. $\Leftarrow$: I want to find the multiplication and…
Wing
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Fundamental group of $\mathcal{B}^2/\mathcal{R}$

Let $\mathcal{B}^2$ be the unit closed disc in $\Bbb{R^2}.$ Now suppose we form the quotient $$\mathcal{B}^2/\mathcal{R}$$ where $\mathcal{R}$ is the equivalence relation generated by $z\sim z\exp{2i\pi /n}$ for $z\in S^1.$ How can I compute the…
Alex
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Understanding orientation of simplicial complex

I am reading a basic course in homology theory from book of paul alexandroff.How can one prove the following:"a closed surface is orientable if and only if one can orient triangles of any of it's triangulations in such a way that the oriented…
Abhishek Shrivastava
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Confused by Hatchers proof of Corollary 2.24

Corollary 2.24 says: If the CW complex $X$ is the union of subcomplexes $A$ and $B$, then the inclusion $(B,A \cap B) \rightarrow (X,A)$ induces isomorphisms $H_{n}(B,A \cap B) \rightarrow H_{n}(X,A)$ for all $n$. Hatcher then gives a very brief…
Tuo
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