Mixing
Uniqueness In Proof For Fundamental Group Of $S^1$
Perhaps a dumb question, but in the standard proof that $pi_1(S^1) cong mathbb{Z}$ (such as the one given in Hatcher), we prove that every loop $gamma$ in $S^1$ is homotopic to some loop in $S^1$ of the form $omega_n(s) = e^{2pi i n s}$, for $s in [0,1]$ and $n in mathbb{Z}$. But then we prove that $n in mathbb{Z}$ is uniquely determined by $gamma$ and hence there is a unique such $omega_n$ to which $gamma$ is homotopic to. Why do we need to prove uniqueness?
general-topology algebraic-topology fundamental-groups
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
add a comment |
Perhaps a dumb question, but in the standard proof that $pi_1(S^1) cong mathbb{Z}$ (such as the one given in Hatcher), we prove that every loop $gamma$ in $S^1$ is homotopic to some loop in $S^1$ of the form $omega_n(s) = e^{2pi i n s}$, for $s in [0,1]$ and $n in mathbb{Z}$. But then we prove that $n in mathbb{Z}$ is uniquely determined by $gamma$ and hence there is a unique such $omega_n$ to which $gamma$ is homotopic to. Why do we need to prove uniqueness?
general-topology algebraic-topology fundamental-groups
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
add a comment |
Perhaps a dumb question, but in the standard proof that $pi_1(S^1) cong mathbb{Z}$ (such as the one given in Hatcher), we prove that every loop $gamma$ in $S^1$ is homotopic to some loop in $S^1$ of the form $omega_n(s) = e^{2pi i n s}$, for $s in [0,1]$ and $n in mathbb{Z}$. But then we prove that $n in mathbb{Z}$ is uniquely determined by $gamma$ and hence there is a unique such $omega_n$ to which $gamma$ is homotopic to. Why do we need to prove uniqueness?
general-topology algebraic-topology fundamental-groups
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
Perhaps a dumb question, but in the standard proof that $pi_1(S^1) cong mathbb{Z}$ (such as the one given in Hatcher), we prove that every loop $gamma$ in $S^1$ is homotopic to some loop in $S^1$ of the form $omega_n(s) = e^{2pi i n s}$, for $s in [0,1]$ and $n in mathbb{Z}$. But then we prove that $n in mathbb{Z}$ is uniquely determined by $gamma$ and hence there is a unique such $omega_n$ to which $gamma$ is homotopic to. Why do we need to prove uniqueness?
general-topology algebraic-topology fundamental-groups
general-topology algebraic-topology fundamental-groups
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
asked Nov 22 '18 at 5:55
Frederic ChopinFrederic Chopin
321111
321111
add a comment |
add a comment |
1 Answer
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Otherwise the fundamental group could be any homomorphic image of $(mathbb{Z}, +)$.
Consider the function that maps every $omega_n$ to its equivalence class of loops.
This is a surjective homomorphism from the group ${omega_n mid nin mathbb N}cong (mathbb{Z}, +)$ to the fundamental group. By showing unicity, you obtain that it is also injective, which yields that the fundamental group be isomorphic to $(mathbb{Z}, +)$.
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
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1 Answer
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Otherwise the fundamental group could be any homomorphic image of $(mathbb{Z}, +)$.
Consider the function that maps every $omega_n$ to its equivalence class of loops.
This is a surjective homomorphism from the group ${omega_n mid nin mathbb N}cong (mathbb{Z}, +)$ to the fundamental group. By showing unicity, you obtain that it is also injective, which yields that the fundamental group be isomorphic to $(mathbb{Z}, +)$.
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
add a comment |
Otherwise the fundamental group could be any homomorphic image of $(mathbb{Z}, +)$.
Consider the function that maps every $omega_n$ to its equivalence class of loops.
This is a surjective homomorphism from the group ${omega_n mid nin mathbb N}cong (mathbb{Z}, +)$ to the fundamental group. By showing unicity, you obtain that it is also injective, which yields that the fundamental group be isomorphic to $(mathbb{Z}, +)$.
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
add a comment |
Otherwise the fundamental group could be any homomorphic image of $(mathbb{Z}, +)$.
Consider the function that maps every $omega_n$ to its equivalence class of loops.
This is a surjective homomorphism from the group ${omega_n mid nin mathbb N}cong (mathbb{Z}, +)$ to the fundamental group. By showing unicity, you obtain that it is also injective, which yields that the fundamental group be isomorphic to $(mathbb{Z}, +)$.
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
Otherwise the fundamental group could be any homomorphic image of $(mathbb{Z}, +)$.
Consider the function that maps every $omega_n$ to its equivalence class of loops.
This is a surjective homomorphism from the group ${omega_n mid nin mathbb N}cong (mathbb{Z}, +)$ to the fundamental group. By showing unicity, you obtain that it is also injective, which yields that the fundamental group be isomorphic to $(mathbb{Z}, +)$.
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
answered Nov 22 '18 at 6:02
A. PongráczA. Pongrácz
5,8881929
5,8881929
add a comment |
add a comment |
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