Proving an “inevitable intersection”
So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that curve are intersected by the curve created by $H$ for some value of $t in [0,1]$. So far, I have tried to prove this by demonstrating that the function that maps the intersection of all curves $H(x,t)$ to any plane within the curves is continuous, but so far said approach has not helped. Could you please tell me how to prove this?
geometric-topology
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
add a comment |
So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that curve are intersected by the curve created by $H$ for some value of $t in [0,1]$. So far, I have tried to prove this by demonstrating that the function that maps the intersection of all curves $H(x,t)$ to any plane within the curves is continuous, but so far said approach has not helped. Could you please tell me how to prove this?
geometric-topology
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
Can you highlight what exactly the question is..
– Praphulla Koushik
2 days ago
Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C times [0,1] rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t in [0,1]$
– Aryaman Gupta
2 days ago
Can you add that I the question.. your $Y$ here is singleton.. Right?
– Praphulla Koushik
2 days ago
add a comment |
So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that curve are intersected by the curve created by $H$ for some value of $t in [0,1]$. So far, I have tried to prove this by demonstrating that the function that maps the intersection of all curves $H(x,t)$ to any plane within the curves is continuous, but so far said approach has not helped. Could you please tell me how to prove this?
geometric-topology
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
So, I need to prove that if a curve $C$ is homotopic to a point (with homotopy $H$ where deformation happens exclusively in the same number of dimensions as $C$), then all of the points within that curve are intersected by the curve created by $H$ for some value of $t in [0,1]$. So far, I have tried to prove this by demonstrating that the function that maps the intersection of all curves $H(x,t)$ to any plane within the curves is continuous, but so far said approach has not helped. Could you please tell me how to prove this?
geometric-topology
geometric-topology
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
asked 2 days ago
Aryaman GuptaAryaman Gupta
336
336
Can you highlight what exactly the question is..
– Praphulla Koushik
2 days ago
Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C times [0,1] rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t in [0,1]$
– Aryaman Gupta
2 days ago
Can you add that I the question.. your $Y$ here is singleton.. Right?
– Praphulla Koushik
2 days ago
add a comment |
Can you highlight what exactly the question is..
– Praphulla Koushik
2 days ago
Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C times [0,1] rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t in [0,1]$
– Aryaman Gupta
2 days ago
Can you add that I the question.. your $Y$ here is singleton.. Right?
– Praphulla Koushik
2 days ago
Can you highlight what exactly the question is..
– Praphulla Koushik
2 days ago
Can you highlight what exactly the question is..
– Praphulla Koushik
2 days ago
Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C times [0,1] rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t in [0,1]$
– Aryaman Gupta
2 days ago
Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C times [0,1] rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t in [0,1]$
– Aryaman Gupta
2 days ago
Can you add that I the question.. your $Y$ here is singleton.. Right?
– Praphulla Koushik
2 days ago
Can you add that I the question.. your $Y$ here is singleton.. Right?
– Praphulla Koushik
2 days ago
add a comment |
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Can you highlight what exactly the question is..
– Praphulla Koushik
2 days ago
Ok, so what I am asking is that if, say, on a plane, there is a contraction (i.e a Homotopy H: C times [0,1] rightarrow Y) of a curve to a point, then how can it be proven that all of the points within the curve must lie on the curve created by $H$ for some $t in [0,1]$
– Aryaman Gupta
2 days ago
Can you add that I the question.. your $Y$ here is singleton.. Right?
– Praphulla Koushik
2 days ago