# Tagged Questions

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### The center of the fundamental group of closed surface [duplicate]

$S^g$ is a closed surface with genus $g$, we know that the fundamental group $\pi_1(S^g)=\{a_1,a_2,\dots ,a_g,b_1,\dots,b_g|a_1b_1a_1^{-1}b_1^{-1}\dots a_gb_ga_g^{-1}b_g^{-1}=1\}$, how to calculate ...
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### Coset of the Abstract index group of a Banach Algebra?

I'm studying on the book of Douglas: "Banach algebra techniques in operator theory" and there is a passage I don't understand, and I hope you can give me a hand. "A continuous function $f$ from $X$ ...
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### Meaning of Fundamental group of a graph

I am a computer science student working in graph algorithms. I am unable to understand what the fundamental group of a graph means. I have some intuition regarding the fundamental group of a ...
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### Fundamental group of $Spin^c(2)$?

Is the fundamental group of $Spin^c(2)$, the second complex spin group, also $\mathbb{Z}$? If so, how does one see this? Just to avoid any confusion, my definition is: Spin^c(2) = (SO(2) \times ...
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### Pure braid group, stabilizer

From group theory we know that a homomorphism $\phi: G \to \operatorname{Sym}(S)$, where S is a set, then $\operatorname{Sym}(S) \cong \Sigma_n$. Its kernel is given as $\bigcap_{s \in S}G_s$, which ...
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### Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?

This question has been cross posted on MathOverflow with some very interesting answers and discussion. I'm currently writing a project on the braid groups and their analogues on closed surfaces. ...
### $X\!\supseteq\!K\!\simeq\!0\Rightarrow X\!\simeq\!X/K$ ($\pi_1$ of a connected graph is free)
How can I prove the following: If $X\supseteq K$ is contractible, then the quotient $X/K$ is homotopy equivalent to $X$? Since $K$ is contractible, we have a homotopy $H:id_K\!\simeq\!c_{k_0}$ ...