Proving surjectivity of Hopf map and existence of Hopf Circles on $mathbb{S}^3$
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I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
asked Nov 25 at 22:04
mathnoob123
688417
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I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
asked Nov 25 at 22:04
mathnoob123
688417
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
– Brian Shin
Nov 26 at 0:57
@BrianShin How can I demonstrate the equivalence of the two maps?
– mathnoob123
Nov 26 at 8:09
add a comment |
up vote
0
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up vote
0
down vote
favorite
I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
asked Nov 25 at 22:04
mathnoob123
688417
I am using the Hopf Map $f: mathbb{S}^3 rightarrow mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q in mathbb{S}^2$ is a circle in $mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.
differential-geometry
differential-geometry
asked Nov 25 at 22:04
mathnoob123
688417
asked Nov 25 at 22:04
mathnoob123
688417
edited Nov 25 at 22:58
asked Nov 25 at 22:04
mathnoob123
688417
asked Nov 25 at 22:04
mathnoob123
688417
asked Nov 25 at 22:04
mathnoob123
688417
688417
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
– Brian Shin
Nov 26 at 0:57
@BrianShin How can I demonstrate the equivalence of the two maps?
– mathnoob123
Nov 26 at 8:09
add a comment |
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
– Brian Shin
Nov 26 at 0:57
@BrianShin How can I demonstrate the equivalence of the two maps?
– mathnoob123
Nov 26 at 8:09
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
– Brian Shin
Nov 26 at 0:57
It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
– Brian Shin
Nov 26 at 0:57
@BrianShin How can I demonstrate the equivalence of the two maps?
– mathnoob123
Nov 26 at 8:09
@BrianShin How can I demonstrate the equivalence of the two maps?
– mathnoob123
Nov 26 at 8:09
add a comment |
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It might be helpful to view the two-sphere and three-sphere in slighty different ways: we can view the two-sphere as $mathbb{C}P^1$, and the three-sphere as the elements in $mathbb{C}^2$ of magnitude 1. Then the Hopf map is given by $(z,w) mapsto [z:w]$.
– Brian Shin
Nov 26 at 0:57
@BrianShin How can I demonstrate the equivalence of the two maps?
– mathnoob123
Nov 26 at 8:09