# Lee: Quotient manifold theorem: smooth transition maps

I'm struggling with the quotient manifold theorem as exposed in John Lee's book "Introduction to smooth manifolds" (theorem 9.16).

This means we can write the transition map between these coordinates as $(\tilde{x},\tilde{x}) = (A(x,y),B(y))$, where $A$ and $B$ are smooth maps defined on some neighborhood of the origin. The transition map $\tilde{\eta} \circ \eta^{-1}$ is just $\tilde{y}= B(y)$, which is clearly smooth.

I don't get why we can write the transition map as such.

PS: The argument is not that long but requires a lot of prelimenary work so I'm not going to post it here. It can be found here: https://www.mathi.uni-heidelberg.de/~lee/StephanSS16.pdf at the end of page 5.

• "I don't understand " is not a question. Your question must be stated in such a way that somebody can answer.. – Thomas Jun 2 '17 at 6:01
• Ok I've added precision on what I don't understand – tomak Jun 2 '17 at 8:40

In the text $(x,y)$ and $(\tilde x, \tilde y)$ are the coordinates for two charts of $M$. Since $M$ has a smooth structure there is a smooth transition map $(\tilde x, \tilde y) = (A(x, y), B(x,y))$ for some smooth functions $A$ and $B$. Then from the argument in the text it follows that $B(x,y) = B(x', y)$ for all $x, x'$, so $B$ can be considered to be a function only of $y$ (if you want to be explicit let $B(y) = B(x_0, y)$, for some arbitrary $x_0$) hence $(\tilde x, \tilde y) = (A(x,y), B(y))$ for some smooth functions $A$ and $B$.