Questions tagged [fundamental-groups]

For questions about or involving the fundamental group.

946 questions
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Etale fundamental group of $Spec \mathbb Z_p[x,y]/(xy-p)$

One can compute etale fundamental group of complex algebraic varieties using topological fundamental group, and we know etale fundamental group of $\mathbb Z_p$ is the same as $\mathbb F_p$. How ...
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Writing every homomorphism $\Upsilon : \pi_{1}(S^1,1) \to \pi_{1}(S^1,1)$ as $\Upsilon = f_\ast.$

I'm proposed the following problem: Show that every group homomorphism $\Upsilon : \pi_1(S^1,1) \to \pi_1(S^1,1)$ can be written as $\Upsilon = f_\ast$ where $f : S^1 \to S^1$ is a continuous map. ...
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Bundle over circle; Monodromy action on cohomology of fiber

I want to understand the following sentence: Let $\pi: M \to S^1$ be a fiber bundle with path connected fiber $F$. Then its monodromy action on $H^k(F; \mathbb C)$ satisfies... How is the ...
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The fundamental group of the gluing of two genus $g$ $3$-dimensional handlebodies

I was given the problem above, and I'd appreciate some help. I think I have a general direction, but I'm not entirely sure if what I'm doing is true, so it'd be great if someone could tell me if what ...
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Calculate the quotient space of three real projective planes

Consider the following quotient space: Take $a,b \in RP^2$, $Y = RP^2 \times \{1, 2, 3\}$, and $X$ = the quotient space of $Y$ quotient by $(b,1) = (a,2)$, $(b,2) = (a,3)$, $(b,3) = (a,1)$. How do ...
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Fundamental group of the real projective plane and its universal cover

I have some questions regarding the real projective plane $\mathbb{R}\mathbb{P}^{2}$. If we choose to represent it using the following identification on the square $[0, 1]^{2}$ : how can we see that ...
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CW complex with fundamental group $\Bbb Z/n$

I'm just learning about CW complexes. I came across this answer: "One way of constructing a connected $2$-dimensional CW complex with fundamental group $\Bbb Z/n$ (for some integer $n\geq 2)$ is to ...
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What would the fundamental group of disjoint union look like?

Fundamental group of a wedge sum is a free product of fundamental groups. Hence, $\pi_1$ maps a coproduct of topological spaces with base points to a coproduct of groups. Since disjoint union(...
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The notation for fundamental group!

I am new in algebraic topology and I am wondering why the notation for the fundamental group is $\pi_1$? I mean what is this "1" for? I searched on the Web and did not find anything! Any idea? Also ...
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Practicing Seifert van Kampen

I am just practicing how to use Seifert-van Kampen. The following excercise is from Hatcher, p.53. Let $X$ be the quotient space of $S^2$ obtained by identifying the north and south poles to a ...
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Local system associated to monodromy representation

How can I associate a local system to a representation $\rho: \pi_1(X) \to \mathbb C^*$? I have seen some construction, but it doesn't click for me. I know that the idea is to use a diagonal action ...
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What is the fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ [duplicate]

I want to know about fundamental group of $RP^{2}$ # $\cdots$ # $RP^{2}$ by Seifert-Van Kampen theorem. In my guessing, that is $\langle a_1, a_2 ,... a_n | a_1^{2}a_2^{2}\cdots a_n^{2}=1\rangle$. ...
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Classification of compact connected manifolds by fundamental group [duplicate]

Every compact connected 2-manifold (I define this as a surface) is homeomorphic to a 2-sphere, a connected sum of tori or a connected sum of projective planes. Since the fundamental groups of the ...
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Automorphism covering space and universal cover

I have again a new question about the automorphism of covering space and the universal cover of a topological space $B$. Actually, let $p : X \rightarrow B$ the universal cover of $B$. I take a normal ...
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How can I visualise the fundamental group of the projective plane?

The real projective plane $\mathbb{RP}^2$ has fundamental group $C_2$. We can understand this via the universal covering mapping $S^2 \to \mathbb{RP}^2$ which identifies antipodal points: the ...
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Let $f : B^2 \rightarrow \mathbb{R}^2$ a continuous map, is $B^2 \subset \operatorname{im}(f)$?

Consider a continuous map $f : B^2 \rightarrow \mathbb{R}^2$ such that $f(S^1) \subset S^1$ and $deg(f_{|S^1}) \ne 0.$ Prove that $B^2 \subset \operatorname{im}(f).$ [Note: Here, $B^2$ is the ...
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Problem with set of cycles covering a graph

My question is the following: is there always, for any non-directed graph $G$, a choice of generators of the fundamental group or, more in general, a set of cycles, covering the graph and with the ...
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Fundamental group of complement of circle with a line passing through it.

Let $X= S^1 \cup y$-axis in the $xy$-plane. Compute the fundamental group of $\mathbb{R}^3 - X$ Edit: To clarify, $S^1 =\{(x,y)|x^2 + y^2 = 1\}$ is the unit circle in the $xy$-plane, so that the $y$...
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Possible groups for fundamental group of $\mathbb{R}$

I want to know if there exist a topology on $\mathbb{R}$ such that $\pi_1(\mathbb{R})\ne (0)$? If not, we conclude that the fundamental group is a property of the underground set, not the topology it ...
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If $A$ is a deformation retract of $X$ and $B$ is a deformation retract of $A$ then $B$ is a deforemation retract of X

If $A$ is a deformation retract of $X$ and $B$ is a deformation retract of $A$ then $B$ is a deformation retract of $X$. I am a beginner in Algebraic Topology so I tried to write every proof out ...
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Fundamental group of tetrahedron

Consider the following double tetrahedron We glue $DCE$ to $CBA$, $CBE$ to $BDA$ and $BDE$ to $DCA$. We call the resulting space $L$. I want to find a cell-structure on $L$ with only two $0$-cells ...
Let $X$ and $Y$ be two topological spaces both with the following cell structure 2 0-cells 4 1-cells 3 2-cells 1 3-cell Can I conclude anything about the fundamental groups of $X$ and $Y$? Are ...