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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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Why is the graph $K_{3,3} $ not one skeleton of the sphere?

I know that someone has already asked the same question here, but there is no solution for part two of the question. And I'm interested in the second part. Here the question: Suppose we build $S^2$ from a finite collection of polygons by…
Rungo
  • 490
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Generators of group $\pi_1(n\mathbb{R}P^2)$

Task: Prove that the group $\pi_1(n\mathbb{R}P^2)$ is generated by elements $a_1,\ldots, a_n,$ obeying unique relation $a^{2}_{1}\cdot\ldots\cdot a^{2}_{n}=1$. I know how to solve it, but I think that $$a^{2}_{1}=1,$$ $$\vdots$$ $$a^{2}_{n}=1.$$ Am…
Aspirin
  • 5,659
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Two deformation retractions (onto $A$) are homotopic (rel $A$).

This is a question from Hatcher's Algebraic Topology (Chapter 0, Question 13): 13. Show that any two deformation retractions $r^0_t$ and $r^1_t$ of a space $X$ onto a subspace $A$ can be joined by a continuous family of deformation retractions…
breeden
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mayer vietoris homework

Let $X=A\cup B\ \ A,B$ are open and $A\cap B$ is contractible. Prove that $H_i(A\cup B)\equiv H_i(A)\oplus H_i(B)$ for $i\geq 2$. I think about using Mayer Vietoris sequence but I don't know how to prove this. Unfortunately, I was off when my class…
hung tran
  • 85
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The homology group of the projective space of dimension $2$

I am reading a book on homological algebra. In order to determine the homology group of the $2$-dimensional projective space, the author identifies the space with the southern hemisphere of $S^2$, together with $s/\{ \pm 1 \}$, $s$ denoting the…
ShinyaSakai
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Classification of circle bundles over a 2-manifold with boundary

I want to understand and try to give a proof of the following claim: Let $B$ be a compact, connected topological $2$-manifold (surface) with nonempty boundary, then the $S^1$-bundles over $B$ with structure group $O(2)$ are in 1-1 correspondence…
Sak
  • 3,866
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Prove $S_g$ retracts to $C$

Let $S_g$ be the closed oriented surface of genus g. Let $C$ be a simple closed curve in $S_g$. Prove that $S_g$ retracts to $C$ if and only if $C$ does not separate $S_g$. If $S_g$ retracts to $C$, by Alexander-Lefschetz duality, I can get…
WWK
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Proof that inclusion in disconnected space cannot be null-homotopic

This is a question related to a question I asked here. I've been thinking about how to prove that if $A \subset X$ where $A$ and $X$ are disconnected spaces and $A$ is a (strong) deformation retract of $X$ and $i: A \hookrightarrow X$ is the…
Rudy the Reindeer
  • 46,359
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Is the dual bundle of canonical line bundle on $\mathbb{CP}^n$ isomorphic to itself?

Is the dual bundle of canonical line bundle on $\mathbb{CP}^n$ isomorphic to itself? The canonical line bundle is represented by $\{g_{ij}\}=\{z_jz_i^{-1}\}$, where $g_{ij}$ is the transformation from $U_j$ to $U_i$ on $U_i\cap U_j$. How to…
user93417
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2 answers

$\mathbb{R}P^1$ is the unit circle $S^1$

The space $P^1$ and the covering map $p:S^1\rightarrow P^1$ are familiar ones. What are they? $S^1=\{z \in \mathbb{C}\mid |z|=1\}$ and $P^1$ is the quotient space where we take $S^1$ and identify $z \in S^1$ with $-z \in S^1$. In a previous…
user43138
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Brieskorn spheres that are S^3

Given pairwise coprime integers $(p,q,r)$, we define a Brieskorn homology 3-spheres by $\Sigma(p,q,r)=\{(z_1,z_2,z_3)\in\mathbb{C}^3 | z_1^p+z_2^q+z_3^r=0, |z_1|^2+|z_2|^2+|z_3|^2=1\}$. Can someone help me show that if say $p=1$, then…
TJIF
  • 463
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1 answer

Hatcher 2.2.26 Show that if $A$ is contractible in $X$ then $H_n(X,A) \approx \tilde H_n(X) \oplus \tilde H_{n-1}(A)$

Show that if $A$ is contractible in $X$ then $H_n(X,A) \approx \tilde H_n(X) \oplus \tilde H_{n-1}(A)$ I know that $\tilde H_n(X \cup CA) \approx H_n(X \cup CA, CA) \approx H_n(X,A)$. And $(X \cup CA)/X = SA$, where $SA$ is the suspension of $A$.…
1LiterTears
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Principal G-bundles depend only on the homotopy type of $G$

Let $G$ be a topological group. Then the classifying space $BG$'s homotopy type depends on the "homotopy type" of the topological group $G$: that is, if $G \to G'$ is a morphism of topological groups (i.e., a continuous homomorphism) which is a weak…
Akhil Mathew
  • 31,310
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1 answer

False proof of $H_0 ( X) = 0$

I've written the following "proof" that if $X$ is path-connected then $H_0 (X) = 0$. I know that that's not the case, yet I can't find the mistake in my "proof". Can you please point it out to me? Here is my "proof": $X$ path-connected $\implies $…
Rudy the Reindeer
  • 46,359
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2 answers

Understanding attaching space

I have some problem understanding the attaching space when learning topology. I cannot understand these two examples. Example 1 Let $X=A=S^1$,$Y=I\times S^1$ and $B=\{0\}\times S^1$. Let $h:B\to A$ be the map that sends $B$ twice around $A$, i.e.,…
Golbez
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