A question about the homotopy. The 2019 Stack Overflow Developer Survey Results Are InA little question about homotopy equivalenceHomotopy equivalence an retractionsHomotopy for the complement when slightly thickening a subspace of $ℝ^n$Homology and Homotopy in the Plane IIQuestion about homotopic functions and homotopy classesProve that there is only one homotopy class of continuous functions from $X$ to $D$Homotopy 'diagrams' for Klein bottle and projective planeTwo questions about homotopy of pathsHomotopy ExerciseFree Homotopy of Loops
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A question about the homotopy.
The 2019 Stack Overflow Developer Survey Results Are InA little question about homotopy equivalenceHomotopy equivalence an retractionsHomotopy for the complement when slightly thickening a subspace of $ℝ^n$Homology and Homotopy in the Plane IIQuestion about homotopic functions and homotopy classesProve that there is only one homotopy class of continuous functions from $X$ to $D$Homotopy 'diagrams' for Klein bottle and projective planeTwo questions about homotopy of pathsHomotopy ExerciseFree Homotopy of Loops
$begingroup$
Recently, I learned the definition of homotopy, and had a question. In all continuous curves(in complex plane) with $a$ as the starting point and $b$ as the end point, if the two curves are homotopic are regarded as the same curve, how many such curves are there?
I would be very happy if someone could answer my question, since this is my first question.
Thank you very much!
general-topology algebraic-topology homotopy-theory
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
$endgroup$
add a comment |
$begingroup$
Recently, I learned the definition of homotopy, and had a question. In all continuous curves(in complex plane) with $a$ as the starting point and $b$ as the end point, if the two curves are homotopic are regarded as the same curve, how many such curves are there?
I would be very happy if someone could answer my question, since this is my first question.
Thank you very much!
general-topology algebraic-topology homotopy-theory
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
$endgroup$
add a comment |
$begingroup$
Recently, I learned the definition of homotopy, and had a question. In all continuous curves(in complex plane) with $a$ as the starting point and $b$ as the end point, if the two curves are homotopic are regarded as the same curve, how many such curves are there?
I would be very happy if someone could answer my question, since this is my first question.
Thank you very much!
general-topology algebraic-topology homotopy-theory
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
$endgroup$
Recently, I learned the definition of homotopy, and had a question. In all continuous curves(in complex plane) with $a$ as the starting point and $b$ as the end point, if the two curves are homotopic are regarded as the same curve, how many such curves are there?
I would be very happy if someone could answer my question, since this is my first question.
Thank you very much!
general-topology algebraic-topology homotopy-theory
general-topology algebraic-topology homotopy-theory
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
edited Mar 23 at 4:58
Thomas Shelby
4,7362727
4,7362727
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
asked Mar 23 at 4:12
Junjie YaoJunjie Yao
1
1
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
There would be only one curve. Suppose $f$ and $g$ are two paths with the same initial and final points. Then define $H:Itimes Ito Bbb C$ by $H(x,t)=(1-t)f(x)+tg(x)$, where $I=[0,1]$. Clearly, $H(x,0)=f(x), H(x,1)=g(x), H(0,t)=f(0)=g(0), H(1,t)=f(1)=g(1)$. Can you check the continuity of $H$?
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
$endgroup$
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
add a comment |
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1 Answer
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1 Answer
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$begingroup$
There would be only one curve. Suppose $f$ and $g$ are two paths with the same initial and final points. Then define $H:Itimes Ito Bbb C$ by $H(x,t)=(1-t)f(x)+tg(x)$, where $I=[0,1]$. Clearly, $H(x,0)=f(x), H(x,1)=g(x), H(0,t)=f(0)=g(0), H(1,t)=f(1)=g(1)$. Can you check the continuity of $H$?
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
$endgroup$
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
add a comment |
$begingroup$
There would be only one curve. Suppose $f$ and $g$ are two paths with the same initial and final points. Then define $H:Itimes Ito Bbb C$ by $H(x,t)=(1-t)f(x)+tg(x)$, where $I=[0,1]$. Clearly, $H(x,0)=f(x), H(x,1)=g(x), H(0,t)=f(0)=g(0), H(1,t)=f(1)=g(1)$. Can you check the continuity of $H$?
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
$endgroup$
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
add a comment |
$begingroup$
There would be only one curve. Suppose $f$ and $g$ are two paths with the same initial and final points. Then define $H:Itimes Ito Bbb C$ by $H(x,t)=(1-t)f(x)+tg(x)$, where $I=[0,1]$. Clearly, $H(x,0)=f(x), H(x,1)=g(x), H(0,t)=f(0)=g(0), H(1,t)=f(1)=g(1)$. Can you check the continuity of $H$?
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
$endgroup$
There would be only one curve. Suppose $f$ and $g$ are two paths with the same initial and final points. Then define $H:Itimes Ito Bbb C$ by $H(x,t)=(1-t)f(x)+tg(x)$, where $I=[0,1]$. Clearly, $H(x,0)=f(x), H(x,1)=g(x), H(0,t)=f(0)=g(0), H(1,t)=f(1)=g(1)$. Can you check the continuity of $H$?
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
answered Mar 23 at 4:50
Thomas ShelbyThomas Shelby
4,7362727
4,7362727
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
add a comment |
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
$begingroup$
The above hint actually proves a stronger notion, called the path homotopy. Actually, any two continuous maps from a space $X$ to $Bbb C$ are homotopic.
$endgroup$
– Thomas Shelby
Mar 23 at 5:03
add a comment |
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