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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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Exerxise 3, chapter 4 of algebraic topology by Hatcher, Page 358

I am stuck on the following problem: For an H-space $(X, x_0)$ with multiplication $\mu: X \times X \to X$, show that the group operation in $\pi_n(X, x_0)$ can also be defined by the rule $(f+g)(x)= \mu(f(x), g(x))$. I have shown that this a…
Amit k
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Finding a covering space for $P \times P$

Let $P$ be a real projective plane. Since the fundamental group of $P \times P$ is $Z_2 \times Z_2$ (abelian group with 5 subgroups), there exists five covering spaces. What is the explicit covering corresponding for $<(1,1)>$? (I found all other 4…
Gobi
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How to prove $S^n \vee S^1$ is not retract of $S^n \times S^1$?

How to prove $S^n \vee S^1$ is not retract of $S^n \times S^1$? If $n=1$, it has been solved here. For greater $n$, the fundamental groups seem not to work.
Kirby Lee
  • 606
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Real projection spaces

It seems that both $RP^2$ and $RP^3$ have the same fundamental group $Z_2$, but Why no map from $RP^3 \to RP^2$ induces an isomorphism between their fundamental groups?
John0417
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Van Kampen Theorem shows $\pi_1(S^1 \vee S^1)\cong \mathbb{Z} * \mathbb{Z}$ and $\pi_1(S^n) \cong \left\{1\right\}$ for $n\geq 2$.

I do not see how using Van Kampen theorem we obtain: (a) $\pi_1(S^1 \vee S^1)\cong \mathbb{Z} * \mathbb{Z}$ and (b) $\pi_1(S^n) \cong \left\{1\right\}$ for $n\geq 2$. In my lecture notes I have that Van Kampen is the main theorem that helps us…
Dragonite
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Deformation Retract is Path Connected

Here is the problem: Let $X \subset Y$ be topological spaces with $X$ a deformation retract of $Y$. Then $Y$ is path connected if $X$ is path connected. This seems simple, but I'm having trouble working out a proof. Here is what I'm thinking…
Ben Tighe
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Homology of the quotient space $(\mathbb{R}^n \times \mathbb{R}^n \setminus \Delta) / (\mathbb{Z} / 2)$

I do not have any clue to the following question: What is the homology group of $(\mathbb{R}^n \times \mathbb{R}^n \setminus \Delta) / (\mathbb{Z} / 2)$, where $\Delta$ is the diagonal in $\mathbb{R}^n \times \mathbb{R}^n$, and the equivalence…
Riley Zhang
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$\pi_0(X) \cong [*,X]$

Let $\pi_0(X)$ denote the set of path components of a space X. I want to show that $\pi_0(X)$ is the set of homotopy classes of maps from the one point space ∗ to X, by constructing a bijection $\pi_0(X) \cong [*,X]$ but I am unsure of where to…
Munkres
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How can be computed $H_n(T_i \times T_j)$?

Here $T_g$ denotes the g-fold torus, and $T_i\times T_j$ denotes the cartesian product. My major trouble is that I can't use Kunneth's Theorem. Thanks for helping!
Edward
  • 21
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How to prove $\Sigma X \cong S^{1}\wedge X$

$\Sigma X$ is the reduced suspension of X, where X is a space based. if anyone has a hint or suggestion of how I can show this result, I would greatly appreciate it
AmottX
  • 407
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Euler Allocation Method

Herr Euler is so prolific that perhaps this has not been asked before. I read a paper - arxiv.org/pdf/0708.2542.pdf "Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle Dirk Tasche" - showing a proof of the "EULER…
rupert
  • 77
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Lemma in the Homotopy Axiom theorem

I have the following Lemma (from Rotman) Let $X$ be a space and, for $i=0,1$, let $\lambda_i^X:X \to X \times I$ be defined by $x \mapsto (x,i)$. If $H_n \left(\lambda_0^X\right)=H_n\left(\lambda_1^X\right):H_n(x)\to H_n \left(X \times I\right)$,…
Juan S
  • 10,268
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Circle Bundles over complex Grassmannian

What is the 'well-known' circle bundle $p:V_{2}(\mathbb{R}^{6})\rightarrow G_{2}(\mathbb{C}^{4})$? Here $V_{2}(\mathbb{R}^{6})$ denotes the real Stiefel manifold of orthonormal 2 frames and $G_{2}(\mathbb{C}^{4})$ is the complex Grassmannian of two…
Bikram Banerjee
  • 41
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Fundamental group of the 3-ball/n-points

Let $D^3 =\{(x,y,z)\in \mathbb{R}^3 :x^2+y^2+z^2 ≤1\}$. Let $A= \{a_1,a_2,...,a_n\} \subset D^3$ be a subset of distinct points in the 3-ball. Compute the fundamental group of the quotient space $\pi_1(D^3/A, b)$ where $b \in D^3/A.$ Should I use…
bmmcutet12
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Van Kampen Theorem Example

The following illustration is given to explain Van Kampen Theorem by the book from Hatcher. In the above example, the line saying "points inside $S^2$ and not in $A$ can be pushed away from $A$ toward $S^2$ or the diameter...". This statement looks…
user3001408
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