- #1
- 7,006
- 10,453
Hi, say X is a topological space with subspaces Y,Z , so that
Y and Z are homotopic in X. Does it follow that there is a continuous
map f:X→X with f(Y)=Z ? Do we need isotopy to guarantee the existence of
a _homeomorphism_ h: X→X , taking Y to Z ?
It seems like the chain of maps parameterized by t , for each x, in
H: XxI→X with H(x,0)=Y and H(x,1)=Z would give us a path H(x,t)
of continuous maps parameterized by t , by composing the maps at each stage t ; from
t=0 to t=1 ; we get an infinite chain of maps, so that
and that the composition H(x,s); s in [0,1]
is a continuous map from X to X taking Y to Z. What if Y,Z are just curves in X ?
Y and Z are homotopic in X. Does it follow that there is a continuous
map f:X→X with f(Y)=Z ? Do we need isotopy to guarantee the existence of
a _homeomorphism_ h: X→X , taking Y to Z ?
It seems like the chain of maps parameterized by t , for each x, in
H: XxI→X with H(x,0)=Y and H(x,1)=Z would give us a path H(x,t)
of continuous maps parameterized by t , by composing the maps at each stage t ; from
t=0 to t=1 ; we get an infinite chain of maps, so that
and that the composition H(x,s); s in [0,1]
is a continuous map from X to X taking Y to Z. What if Y,Z are just curves in X ?