I'm currently studying Knot Theory from Gerhard Burde - Heiner Zieschang "Knots". I'm stuck with the proof of $(1) \Rightarrow 2$ the following:
The question was asked previously here and here, but the theorems used aren't within my reach, I'm trying to find a proof as elementary as possible.
The first question is the definition of simplex star, not given previously in the book. The troubles comes with the omitted details left in the following sentence:
Next, we choose a point $P_1 \ne P_0$ in the simplex star of $P_0$, and by similar arguments we construct an ambienty isotopy $H^1$ with $h_1^1 \circ h_1^0 \circ f$ leaving fixed the $1-$simple $[P_0,P_1]$.
Any help, details or alternative proof of the above proposition would be appreciated. I also left a picture of Figure $1.5$.
