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I think it's accepted to post HW type question in here.
"Is there a submersion from S^1 to R? From R to S^1?"
By a submersion from M to N, I mean a map f:M-->N whose tangent map is surjective.
I answered 'yes' to both questions, which I find dubious.
Take for an atlas of S^1, {({[itex]\exp(i\theta):0<\theta<2\pi[/itex]}, z-->arg(z)), ({[itex]\exp(i\theta):-\pi<\theta<\pi[/itex]}, z-->arg(z))}
Submersion from S^1 to R: Let x be in S^1 and f:S^1-->[0,2pi[ be f(z)=arg(z). Let r be a real number. I must show that there is a path y:]-e,e[-->S^1 such that y(0)=x and d/dt(f o y)(0)=r. Well such a path is y(t)=xexp(irt).
Submersion from R to S^1: Let x to be in R and f:R-->S^1 be f(y)=exp(ig(y)), where g:R-->[0,2pi[ is the "mod 2 pi" map. Let r be a real number. I must show that there is a path y:]-e,e[-->R such that y(0)=x and d/dt(p o f o y)(0)=r, for p a chart of S^1 around f(x). Well such a path is y(t)=x(t+1)^(r/x). I have constructed y so that y'(0)=r.
Yes, because by construction, p o f = g, and d/dt(g o y)(0)=g'(y(0))*y'(0)=1*r=r.
"Is there a submersion from S^1 to R? From R to S^1?"
By a submersion from M to N, I mean a map f:M-->N whose tangent map is surjective.
I answered 'yes' to both questions, which I find dubious.
Take for an atlas of S^1, {({[itex]\exp(i\theta):0<\theta<2\pi[/itex]}, z-->arg(z)), ({[itex]\exp(i\theta):-\pi<\theta<\pi[/itex]}, z-->arg(z))}
Submersion from S^1 to R: Let x be in S^1 and f:S^1-->[0,2pi[ be f(z)=arg(z). Let r be a real number. I must show that there is a path y:]-e,e[-->S^1 such that y(0)=x and d/dt(f o y)(0)=r. Well such a path is y(t)=xexp(irt).
Submersion from R to S^1: Let x to be in R and f:R-->S^1 be f(y)=exp(ig(y)), where g:R-->[0,2pi[ is the "mod 2 pi" map. Let r be a real number. I must show that there is a path y:]-e,e[-->R such that y(0)=x and d/dt(p o f o y)(0)=r, for p a chart of S^1 around f(x). Well such a path is y(t)=x(t+1)^(r/x). I have constructed y so that y'(0)=r.
Yes, because by construction, p o f = g, and d/dt(g o y)(0)=g'(y(0))*y'(0)=1*r=r.