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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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On certain cases of Seifert Van Kampen

Given two connected open sets $U,V \subset X$ such that $U \cap V$ is path connected and $U \cup V = X$, then $\pi_1(U) \ast_{\pi_1(U \cap V)} \pi_1(V) \cong \pi_1(X)$. This is of course the Seifert Van Kampen theorem, a question in Munkres asks if…
JSchlather
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how to calculate the fundamental group using intitutive ideas?

I am trying to calculate the fundamental group of $\mathbb{R}^3- \{ x\text{-axis}\cup y\text{-axis}\cup z\text{-axis}\}$. Idea: I think we can show it deformation retracts on 2-sphere minus 4 points.
hina
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Is a loop in $X$ based at $a\in X$ homotopic to its reverse loop?

A loop in a topological space $X$ based at $a\in X$ is a continuous function $\rho: [0,1]\rightarrow X$ such that $\rho(0)=\rho(1)=a$. The reverse of $\rho$ is $\overline{\rho}$ such that $\overline{\rho}(s)=\rho(1-s)$ for each $s\in [0,1]$.…
Sid Caroline
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Local degree of induced map $\hat{f}$ on Riemann surface for a polynomial $f$

This is an exercise from Allen Hatcher's algebraic topology. The question is: A polynomial $f(z)$ with complex coefficients, viewed as a map $C → C$, can always be extended to a continuous map of one-point compactifications $\hat{f}: S^2 → S^2$. Show…
non-abelian group of order 9
  • 881
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Simply connected covering

Question: Construct a simply connected covering which a subspace of $\mathbb R^3$ of union of a sphere and a circle intersecting in two points. My idea: First of all note that union of a sphere and a circle intersecting in two points is homotopy…
tina
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Finding map from Klein bottle to $RP^2$ that induces an epimorphism of fundamental groups.

I want to find a map from the Klein bottle to $RP^2$ that induces an epimorphism of fundamental groups. Since the fundamental group of $RP^2$ is $Z_2$, intuitively it feels like a good start would be to look for any non-trivial map from $K$ to…
TheMountainThatCodes
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Singular homology of the sphere

By Mayer-Vietoris, to compute the singular homology of the $S^n$, essentially we want to look at the following exact sequence $0 \to H_1(S^1) \xrightarrow{\partial^{\ast}} H_0(S^0) \xrightarrow{k_{\ast}} H_0(B^1) \oplus H_0(B^1)…
Adam
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Homotopy group realization

I am looking for information related to the following question: For which $n \in\mathbb{N}$ can every group $G$ be realized as $\pi_n(X)$ for some space $X$? I have seen in Hatcher that $n=1$ is one such case. I was wondering whether this result…
Holdsworth88
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Killing successive homotopy groups via fibrations

Let $X$ be some sort of sufficiently nice space, e.g. a (connected) cell complex. Then $X$ has a universal cover $\tilde{X}$. This is simply connected by definition and it is easy to show that $\pi_n(\tilde{X})\cong\pi_n(X)$ for all $n>1$. We have,…
Miha Habič
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Why is the Klein bottle not homeomorphic to a subspace of $\mathbb R^3$?

It is stated in may books and also in the wikipedia or WolframMathWorld that the Klein bottle is not homeomorphic to a subspace of $\mathbb R^3$. Why is it so?
Jochen
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Two apparently contradictory statements in Strom's "Modern Classical Homotopy Theory"

I am struggling to see where this apparent contradiction is not a contradiction: these examples and exercise all come from Strom's book "Modern classical algebraic topology": of course the map $[S^1, (S^1\vee S^1)\amalg (S^1\vee S^1)] \to \langle…
fosco
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Softwares for visualizing $2$-dimensional CW complexes

Are there any softwares that help visualize a CW-complex? My input data is a combinatorially defined cell structure on non-orientable surfaces. In particular I know the number of cells (the $2$-cells are regular polygons) and the attaching maps.
Priyavrat Deshpande
  • 205
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Chain maps induces by maps of topological spaces

Given two topological spaces $X,Y$. Is every chain map $f_{\ast}:S_{\ast}(X)\rightarrow S_{\ast}(Y)$ induced by a map (of topological spaces) $f:X\rightarrow Y$?
user372565
4
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1 answer

Covering Space of Circle

It is easy to see that $p:\mathbb{R} \rightarrow S^1: p(t)=(\cos t, \sin t)$ is a covering map of $S^1$ (Indeed take a point $x_0$ in $S^1$, take $U=S^1\setminus \{-x_0\}$ as an open neighborhood and then $p^{-1}(U)=\cup_{n \in Z} J_n$ where…
Juan S
  • 10,268
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Is the 1-dimension projective line a circle?

Is the 1 dimensional projective line homeomorphic to the circle? If so, the circle is homeomorphic to itself with antipodal points identified (very unintuitive). I am missing something?
PossumP
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