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Tom Goodwillie
Tom Goodwillie
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In the topological category the usual compact-open topology does the job. (EDIT: Or rather I suppose you might have to modify it a little so that $h\mapsto h^{-1}$ is continuous.) At times you might want to replace a function space by the simplicial set called its total singular complex. This is an instance of the more general procedure of replacing a space by its total singular complex when convenient, a trend that began around that same time. But there may not be any great technical advantages in doing so when working with spaces of homeomorphisms.

In the $PL$ category it gets worse: I don't know a useful topology on the group of $PL$ automorphisms of $M$. Topologizing it as a subspace of the space of homeomorphisms can't be right, and there isn't something like a Whitney topology to fix things up.

EDIT: This seems about right as a reason why $PL$ homeomorphisms cannot be treated as a subspace of homeomorphisms. Suppose $h$ is a $PL$ automorphism of $\mathbb R^n$ fixing the origin. In a neighborhood of the origin it is radial (linear on each ray). By a sort of reverse Alexander trick you would be able to make it radial everywhere if you were allowed to use isotopies through $PL$ homeomorphisms rather than actual $PL$ isotopies. In this way you would get a deformation retraction to the subgroup consisting of radial $PL$ homeomorphisms. Then you could further deform to the smaller group that consists of those radial homeomorphisms that are made by coning $PL$ homeomorphisms of a ($PL$) $(n-1)$-sphere. That would mean that the canonical map from $Aut^{PL}(D^n)\simeq Aut^{PL}(S^{n-1})$ to $Aut^{PL}(D^n-S^{n-1})$, or to germs at the origin, is an equivalence. But the fiber of this map to germs is the pseudoisotopy space $P^{PL}(S^{n-1})$, and this is not contractible. Contradiction. Why is $P^{PL}(S^{n-1})$ not contractible? It fibers over the pseudoisotopy embedding space $PE^{PL}(D^{n-1},S^{n-1})$, with contractible fiber $P^{PL}(D^{n-1})$. The space $PE^{PL}(D^{n-1},S^{n-1})$ is not contractible because (1) the smooth analogue $PE^{Diff}(D^{n-1},S^{n-1})$ is contractible by Hatcher's light bulb trick, (2) the fiber of $PE^{Diff}(D^{n-1},S^{n-1})\to PE^{PL}(D^{n-1},S^{n-1})$ is equivalent by smoothing theory to $\Omega fiber (O_n/O_{n-1}\to PL_n/PL_{n-1})$, and (3) this last is not contractible. Why (3)? Because if it were contractible then the fiber of the map $P^{Diff}(D^{n-1})\to P^{PL}(D^{n-1})$, which is $\Omega^n fiber (O_n/O_{n-1}\to PL_n/PL_{n-1})$ by smoothing theory, would be contractible. But $P^{PL}(D^{n-1})$ is contractible while $P^{Diff}(D^{n-1})$ is not by Waldhausen (rationally it becomes $\Omega^2 K(\mathbb Z)$ as $n\to \infty$).

In the topological category the usual compact-open topology does the job. At times you might want to replace a function space by the simplicial set called its total singular complex. This is an instance of the more general procedure of replacing a space by its total singular complex when convenient, a trend that began around that same time. But there may not be any great technical advantages in doing so when working with spaces of homeomorphisms.

In the $PL$ category it gets worse: I don't know a useful topology on the group of $PL$ automorphisms of $M$. Topologizing it as a subspace of the space of homeomorphisms can't be right, and there isn't something like a Whitney topology to fix things up.

In the topological category the usual compact-open topology does the job. (EDIT: Or rather I suppose you might have to modify it a little so that $h\mapsto h^{-1}$ is continuous.) At times you might want to replace a function space by the simplicial set called its total singular complex. This is an instance of the more general procedure of replacing a space by its total singular complex when convenient, a trend that began around that same time. But there may not be any great technical advantages in doing so when working with spaces of homeomorphisms.

In the $PL$ category it gets worse: I don't know a useful topology on the group of $PL$ automorphisms of $M$. Topologizing it as a subspace of the space of homeomorphisms can't be right, and there isn't something like a Whitney topology to fix things up.

EDIT: This seems about right as a reason why $PL$ homeomorphisms cannot be treated as a subspace of homeomorphisms. Suppose $h$ is a $PL$ automorphism of $\mathbb R^n$ fixing the origin. In a neighborhood of the origin it is radial (linear on each ray). By a sort of reverse Alexander trick you would be able to make it radial everywhere if you were allowed to use isotopies through $PL$ homeomorphisms rather than actual $PL$ isotopies. In this way you would get a deformation retraction to the subgroup consisting of radial $PL$ homeomorphisms. Then you could further deform to the smaller group that consists of those radial homeomorphisms that are made by coning $PL$ homeomorphisms of a ($PL$) $(n-1)$-sphere. That would mean that the canonical map from $Aut^{PL}(D^n)\simeq Aut^{PL}(S^{n-1})$ to $Aut^{PL}(D^n-S^{n-1})$, or to germs at the origin, is an equivalence. But the fiber of this map to germs is the pseudoisotopy space $P^{PL}(S^{n-1})$, and this is not contractible. Contradiction. Why is $P^{PL}(S^{n-1})$ not contractible? It fibers over the pseudoisotopy embedding space $PE^{PL}(D^{n-1},S^{n-1})$, with contractible fiber $P^{PL}(D^{n-1})$. The space $PE^{PL}(D^{n-1},S^{n-1})$ is not contractible because (1) the smooth analogue $PE^{Diff}(D^{n-1},S^{n-1})$ is contractible by Hatcher's light bulb trick, (2) the fiber of $PE^{Diff}(D^{n-1},S^{n-1})\to PE^{PL}(D^{n-1},S^{n-1})$ is equivalent by smoothing theory to $\Omega fiber (O_n/O_{n-1}\to PL_n/PL_{n-1})$, and (3) this last is not contractible. Why (3)? Because if it were contractible then the fiber of the map $P^{Diff}(D^{n-1})\to P^{PL}(D^{n-1})$, which is $\Omega^n fiber (O_n/O_{n-1}\to PL_n/PL_{n-1})$ by smoothing theory, would be contractible. But $P^{PL}(D^{n-1})$ is contractible while $P^{Diff}(D^{n-1})$ is not by Waldhausen (rationally it becomes $\Omega^2 K(\mathbb Z)$ as $n\to \infty$).

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Tom Goodwillie
Tom Goodwillie
  • 55.9k
  • 7
  • 151
  • 240

In the topological category the usual compact-open topology does the job. At times you might want to replace a function space by the simplicial set called its total singular complex. This is an instance of the more general procedure of replacing a space by its total singular complex when convenient, a trend that began around that same time. But there may not be any great technical advantages in doing so when working with spaces of homeomorphisms.

In the smooth category you can give the space of diffeomorphisms the Whitney topology and then look at the total singular complex, but in some ways it's a better idea to look at the smaller simplicial set consisting of the "smooth" (rather than merely continuous) maps of simplices into the space, meaning those which correspond to smooth maps $\Delta\times M\to M$. Note that when dealing with smooth bundles with fiber $M$ over a base $B$ you really are interested more in "smooth" maps from open subsets $U\subset B$ into the group than in continuous maps, because it is the former that correspond to automorphisms of the trivial bundle $U\times M$. It is true that, just as for maps between finite-dimensional manifolds, smooth homotopy classes of smooth maps from $U$ to the group are in bijection with homotopy classes of continuous maps from $U$ to the group. It is not true that the compact-open topology on smooth automorphisms is just as good as the Whitney topology for these purposes: you don't want to be able to kill an exotic diffeomorphism of $D^n$ that fixes a neighborhood of the boundary sphere by the Alexander trick (which scales it through diffeomorphisms until at the last moment it becomes the identity).

In the $PL$ category it gets worse: I don't know a useful topology on the group of $PL$ automorphisms of $M$. Topologizing it as a subspace of the space of homeomorphisms can't be right, and there isn't something like a Whitney topology to fix things up.

One other point:

What about the structure group for microbundles? You want to compare on the one hand the group of automorphisms of $\mathbb R^n$ fixing the origin (the structure group for open disk bundles with section) and on the other hand the group of invertible germs of maps $\mathbb R^n\to \mathbb R^n$ at the origin (the structure group for microbundles). As I understand the Kister-Mazur Theorem, bundles are "the same" as microbundles because the comparison map of groups is an equivalence. The kernel of the comparison map (i.e. the group of all automorphisms of $\mathbb R^n$ such that some neighborhood of the origin is pointwise fixed) is rather obviously contractible, while on the other hand the map is surjective (every invertible germ is the germ of a global automorphism). I think that in getting the details of that right it is convenient not to have to actually topologize the group of germs, but rather to speak of germs of maps from $\Delta^k\times \mathbb R^n$ to itself along $\Delta^k\times 0$.