Uniqueness In Proof For Fundamental Group Of $S^1$
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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 at 5:55
Frederic Chopin
322111
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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 at 5:55
Frederic Chopin
322111
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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 at 5:55
Frederic Chopin
322111
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 at 5:55
Frederic Chopin
322111
asked Nov 22 at 5:55
Frederic Chopin
322111
asked Nov 22 at 5:55
Frederic Chopin
322111
asked Nov 22 at 5:55
Frederic Chopin
322111
asked Nov 22 at 5:55
Frederic Chopin
322111
322111
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add a comment |
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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 at 6:02
A. Pongrácz
5,178725
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
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 at 6:02
A. Pongrácz
5,178725
add a comment |
up vote
1
down vote
accepted
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 at 6:02
A. Pongrácz
5,178725
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
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 at 6:02
A. Pongrácz
5,178725
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 at 6:02
A. Pongrácz
5,178725
answered Nov 22 at 6:02
A. Pongrácz
5,178725
answered Nov 22 at 6:02
A. Pongrácz
5,178725
answered Nov 22 at 6:02
A. Pongrácz
5,178725
5,178725
add a comment |
add a comment |
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