# Questions tagged [fundamental-groups]

For questions about or involving fundamental groups of topological spaces, as well as related topics such as fundamental groupoids and étale fundamental groups.

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### Bijection Between Maps of Covering Spaces and Maps of Fundamental Groupoid Covers

I'm reading May's Concise Course in Algebraic Topology. He states: My question is about this last corollary. How does the bijection $\text{Cov}(E, E') \to \text{Cov}(\Pi(E), \Pi(E'))$ immediately ...
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### Question on induced homomorphism involving translation maps

I'm working with Viro's textbook on topology, and I stuck on this exercise on induced homomorphisms: Where $T_s : \pi_1(X,x_0) \to \pi_1(X,x_1) : [\alpha] \mapsto [s^{-1}\alpha s]$ -- a translation ...
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### Fundamental group of the total space of an oriented $S^1$ fiber bundle over $T^2$

Let $M$ be the total space of an oriented $S^1$ fiber bundle over $T^2$. Can we show the fundamental group of $M$ is nilpotent? More generally, how can we calculate the fundamental group of $M$ ...
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### Intuition behind coverings of $S^1 \vee S^1$

I was recently studying Algebraic Topology reading Hatcher, and came across a table of diagram that talks about covering spaces of $S^1 \vee S^1$ on page 58. I don't really understand how these are ...
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### Cell structure of a torus with an open disk removed

I'm reviewing Algebraic Topology and this is an old homework "Viewing the torus T as usual as the square $[-1,1]^2$ with opposite sides identified, let $X$ be obtained from T by removing the open ...
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### determining the quotient group in Mayer-Vietoris sequence

I am having trouble to determine the quotient group in the following Mayer-Vietoris sequence. I know this problem in Hatcher exists here but my question is not to have a solution (because I do have ...
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### Action of the Fundamental Group of Klein Bottle over $\mathbb{R}^{2}$

It is well know that the Fundamental Group of the Klein Bottle is defined by $$G=BS(1,-1)=\langle a,b:bab^{-1}=a^{-1}\rangle$$ An explicit description can be obtaned by define $G$ as the group of ...
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### Calculate $p_*(\pi_1(\tilde{X},e_i))$

Give an example of a two-fold cover $(\tilde{X},p)$ of figure eight. For those examples choose a basepoint $e$ and a base point $e_i\in \tilde{X}$ and calculate $p_*(\pi_1(\tilde{X},e_i))$ My attempt: ...
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### Connected components of free loop space

Let $X$ be a topological space. And let $\Lambda X=\mathrm{Top}\left[S^1,X\right]$ be the space of continuous loops in $X$. Then how do we calculate $\Pi_0\Lambda X$, the set of connected components ...
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### For a topological manifold $X$ is it true that $X$ is a covering of $X\lor X$?

Here is my question: Let $X$ be a topological manifold. Is it true that $X$ is a covering of $X\lor X$ and $X\lor X\lor X$ and, so on. I have a intuition, $\pi_1(X\lor X)=\pi_1(X)*\pi_1(X)$.
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### How lemma 2 is a corollary of lemma 1?

We took this lemma 2: If there is a pointed homotopy between $f$ and $g$ where they are functions from $(X, x_{0})$ to $(Y, y_{0})$ then $$f_{*} = g_{*}.$$ And our professor said (without proof) that ...
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### Lifting of homotopy

Let $F: I \times I \rightarrow \mathbb{S}^1$ a continuous function such that $F(0,t)=F(1,t)=p \in \mathbb{S}^1$ for a fixed $p$ (i.e. a homotopy between two loops in the circle based at $p$). Show ...
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### Finding subgroups of $G=\langle x,y,z~|~x^2y^2z^2 \rangle$ using covering space theory

Consider the group $G$ having a presentation $G=\langle x,y,z~|~x^2y^2z^2 \rangle$. I am trying to find all subgroups of $G$ of index 6 using covering space theory. It is well-known that the connected ...
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### How to prove that the complex logarithm is not continuous using the Fundamental Group

I need to prove that the function $f:\mathbb{C^*} \rightarrow \mathbb{C} ; \exp{(f(z))} = z$ is not continuous using the fundamental group. I´ve found this Does every continuous map induce a ...
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### No closed orientable 3-manifold is homotopy equivalent to $S_g\vee S^3$
I am working on ALgebraic Topology in my past Quals. "Prove that no closed orientable 3-manifold is homotopy equivalent to $S_g\vee S^3$, where $S_g$ is the orientable surface of genus \$g\...