It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details. In a way it's more of a ground-up collection of basic information on the topology of function spaces.

Regarding your 2nd question, if instead of demanding a fibre bundle you ask for a Serre fibration, the proof is relatively simple. It's just the isotopy extension theorem with parameters, and the proof is pretty much verbatim Hirsch's proof of isotopy extension in his "Differential Topology" text plus the observation that solutions depend smoothly on the initial conditions.

Regarding your 2nd question, yes of course. Palais's paper is quite nice. If you haven't had a look at it, you might as well try. If you want to discover the proof on your own I'd start with the case $S$ a finite set. Then move up to $S$ a positive-dimensional submanifold. You'll want to be comfortable with things like the proof of the tubular neighbourhood theorem, the concept of injectivity radius, etc.

It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details. In a way it's more of a ground-up collection of basic information on the topology of function spaces.

Regarding your 2nd question, if instead of demanding a fibre bundle you ask for a Serre fibration, the proof is relatively simple. It's just the isotopy extension theorem with parameters, and the proof is pretty much verbatim Hirsch's proof of isotopy extension in his "Differential Topology" text plus the observation that solutions depend smoothly on the initial conditions.

Regarding your 2nd question, yes of course. Palais's paper is quite nice. If you haven't had a look at it, you might as well try -- it's only 7 pages long. If you want to discover the proof on your own I'd start with the case $S$ a finite set. Then move up to $S$ a positive-dimensional submanifold. You'll want to be comfortable with things like the proof of the tubular neighbourhood theorem, the concept of injectivity radius, etc.

It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details although I believe it pre-dates Palais's work.

Regarding your 2nd question, if instead of demanding a fibre bundle you ask for a Serre fibration, the proof is relatively simple. It's just the isotopy extension theorem with parameters, and the proof is pretty much verbatim Hirsch's proof of isotopy extension in his "Differential Topology" text plus the observation that solutions depend smoothly on the initial conditions.

Regarding your 2nd question, yes of course. Palais's paper is quite nice. If you haven't had a look at it, you might as well try. If you want to discover the proof on your own I'd start with the case $S$ a finite set. Then move up to $S$ a positive-dimensional submanifold. You'll want to be comfortable with things like the proof of the tubular neighbourhood theorem, the concept of injectivity radius, etc.

It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details. In a way it's more of a ground-up collection of basic information on the topology of function spaces.

Regarding your 2nd question, if instead of demanding a fibre bundle you ask for a Serre fibration, the proof is relatively simple. It's just the isotopy extension theorem with parameters, and the proof is pretty much verbatim Hirsch's proof of isotopy extension in his "Differential Topology" text plus the observation that solutions depend smoothly on the initial conditions.

Regarding your 2nd question, yes of course. Palais's paper is quite nice. If you haven't had a look at it, you might as well try. If you want to discover the proof on your own I'd start with the case $S$ a finite set. Then move up to $S$ a positive-dimensional submanifold. You'll want to be comfortable with things like the proof of the tubular neighbourhood theorem, the concept of injectivity radius, etc.