3 more examples

Some pretty fundamental objects in manifold theory are Banach manifolds. Perhaps the simplest would be the space of $C^k$ diffeomorphisms of a compact manifold M for $k > 0$ finite. There's of course all kinds of variations on this theme: embedding spaces, diffeomorphisms that preserve a structure of some sort, the space of smooth maps between two manifolds, etc, etc.

So they're used to prove a variety of results, and they are the "targets" of many important theorems. For example, Smale's theorem on the homotopy type of $Diff(S^2)$, or the Smale-Hirsch theorem on the homotopy type of the space of immersions of one manifold in another. Palais's theorem that restriction maps are fiber bundles (more easily seen to be fibrations) was used by Fadell and Neuwirth to show the pure braid groups are iterated semi-direct products of free groups. The fact that $Diff(S^1)$ has the homotopy type of $O_2$ says that a circle bundle over a space is always equivalent to a circle bundle with linear structure group, etc.

IMO the answer to your question can be summed up in a little nugget of an observation that if you value manifolds, it's only natural to value the automorphisms of manifolds, and mapping spaces of manifolds. And since the automorphisms these mapping spaces have a natural structure (of a Banach manifold) certainly that should be relevant.

2 elaboration

Some pretty fundamental objects in manifold theory are Banach manifolds. Perhaps the simplest would be the space of $C^k$ diffeomorphisms of a compact manifold M for $k > 0$ finite. There's of course all kinds of variations on this theme: embedding spaces, diffeomorphisms that preserve a structure of some sort, etc, etc.

So they're used to prove a variety of results, and they are the "targets" of many important theorems. For example, Smale's theorem on the homotopy type of $Diff(S^2)$, or the Smale-Hirsch theorem on the homotopy type of the space of immersions of one manifold in another. Palais's theorem that restriction maps are fiber bundles (more easily seen to be fibrations) was used by Fadell and Neuwirth to show the pure braid groups are iterated semi-direct products of free groups. The fact that $Diff(S^1)$ has the homotopy type of $O_2$ says that a circle bundle over a space is always equivalent to a circle bundle with linear structure group, etc.

IMO the answer to your question can be summed up in a little nugget of an observation that if you value manifolds, it's only natural to value the automorphisms of manifolds. And since the automorphisms have a natural structure (of a Banach manifold) certainly that should be relevant.

1

Some pretty fundamental objects in manifold theory are Banach manifolds. Perhaps the simplest would be the space of $C^k$ diffeomorphisms of a compact manifold M for $k > 0$ finite. There's of course all kinds of variations on this theme: embedding spaces, diffeomorphisms that preserve a structure of some sort, etc, etc.

So they're used to prove a variety of results, and they are the "targets" of many important theorems. For example, Smale's theorem on the homotopy type of $Diff(S^2)$, or the Smale-Hirsch theorem on the homotopy type of the space of immersions of one manifold in another. Palais's theorem that restriction maps are fiber bundles (more easily seen to be fibrations) was used by Fadell and Neuwirth to show the pure braid groups are iterated semi-direct products of free groups.

IMO the answer to your question can be summed up in a little nugget of an observation that if you value manifolds, it's only natural to value the automorphisms of manifolds. And since the automorphisms have a natural structure (of a Banach manifold) certainly that should be relevant.