Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

949 questions
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What does n times a generator of a cyclic group mean?

It seems to be saying that, for some integer deg(f) it sends the generator i to deg(f) times the generator, but I can only see it making sense if it sends it to the generator^deg(f) What does it mean ...
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Deck transformation group isomorphic to $\mathbb{Z}/\mathbb{6Z}$

I'm looking for a topological space $X$ and a covering space $(\tilde X,p)$ such that the deck transformation group Aut$_{X}(\tilde X)$ is isomorphic to $\mathbb{Z}/\mathbb{6Z}$. My idea is to find ...
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Why is $N(p_*(\pi_1(T, \overline{x}_0)))/p_*\pi_1(X, x_0) \cong D$?

The following is from "Homotopical Topology" by Fuchs et al. Let $p:T \to X$ be a covering, $\overline{x}_0 \in T$, $p(\overline{x}_0) = x_0 \in X$, $D$ the group of deck transformations of $T$, and ...
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How to build homotopy from convex to a point?

How to build homotopy from convex to a point? for example I want to build homotopy from the real line $\Bbb R$ to a single point $x_0$ I know that every path in a closed set $[a,x_0]$ in $\Bbb R$ ...
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Equivalence of Categories between the Fundamental Group and Groupoid

Let $X$ be a path connected space, and let $x \in X$. Then we have that $\pi_1 (X,x)$ is a full subcategory of $\Pi(X)$. So, the inclusion functor $J: \pi_1(X,x) \to \Pi(X)$ is an equivalence of ...
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Find fundamental group of space $X=(S^1\times S^1)/(S^1\times \{s_0\})$

Find fundamental group of space $X=(S^1\times S^1)/(S^1\times \{s_0\})$ I am not sure how to do it. It seems obvious that $\pi_1(X)=\mathbb{Z}$ because $X$ is torus where one class of nontrivial ...
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Prove that $f:X\to S^1$ is null-homotopic iff homomorphism $f_*:\pi_1(X,x)\to\pi_1(S^1,f(x))$ is trivial for some $x$.

Let $X$ be connected and locally path connected space. Prove that: $f:X\to S^1$ is null-homotopic iff homomorphism $f_*:\pi_1(X,x)\to\pi_1(S^1,f(x))$ is trivial for some $x$. I do not have any idea ...
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Role of path connectedness in fundamental group of product space

Proposition 1.12 from Hatcher's Algebraic Topology book states that \begin{align*} \pi_1(X \times Y) \; \text{is isomorphic to} \; \pi_1(X) \times \pi_1(Y) \; \text{if X and Y are path connected} \; \...
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Does the Union of simply connected spaces with with certain conditions,simply connected?

Does the Union of finite simply connected open subspaces of a space, with the condition that for each three of these sub spaces the intersection is also simply connected, also have to be simply ...
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Classify, except isomorphism, all the associated covering spaces of $3$ sheets on $\mathbb{S}^1\times\mathbb{S}^1$. Do the same for the

Classify, except isomorphism, all the associated covering spaces of $3$ sheets on $\mathbb{S}^1\times\mathbb{S}^1$. Do the same for the covering spaces of $4$ sheets. I know that $G=\mathbb{Z}^2$ ...
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Let $K$ be the Klein bottle. Give examples of cover spaces $p_1:E_1\to K, p_1:E_2\to K$ each with two sheets and such that $E_1$ and $E_2$

Let $K$ be the Klein bottle. Give examples of cover spaces $p_1:E_1\to K, p_1:E_2\to K$ each with two sheets and such that $E_1$ and $E_2$ are arc-connected but not isomorphic as covering spaces. I ...