This is my first post, and I plan on sporadically writing some in the future. I’m Peter, a third year grad student at Cornell, and I talk to Greg pretty often, so I thought I’d write down some of the things I say. This first post won’t be long or deep, but it’s kind of cute, and the trick behind it is useful in many other situations, so I decided to share it. Let’s say we have a finite group acting on a compact manifold . The Whitney embedding theorem says that we can embed into for sufficiently large , and what I want to show in this post is that you can do this in a -equivariant way, i.e. there is an embedding and an injective homomorphism such that . I guess the moral of the story is that compact manifolds are really just “nice” subsets of Euclidean space, and a compact manifold with a finite group action is really nothing but a “nice” subset of Euclidean space that is preserved by the action of a finite group of permutation matrices. Also, if one were to summarize the moral of the trick used, one might say “averaging over the elements of a group makes things equivariant,” and this idea comes up almost uncountably many times in many areas of mathematics (of course, averaging takes on different meanings in different situations).
The notation for this is a little messy, but the idea isn’t too hard. First we define an embedding and then we show that this embedding is -equivariant. Pick a finite covering of by coordinate charts and a partition of unity which is subordinate to the cover of . To make the notation a little more clear later, we can break up the into their components . Now we let be the standard basis for with , , and . (To simplify things, I’ll use the same bounds for each index letter throughout the post.) We also let be the standard basis for . Now let be the vector space spanned by and . Then we can define our embedding with the following formula:
(In this formula we are defining if even if is undefined.) Since all the functions involved are smooth, this is obviously a smooth map, so to show it is an embedding it is sufficient to show it is a homeomorphism. Since is compact and its image is Hausdorff, it is sufficient to prove that is one-to-one. Let and suppose that . Then since the are a partition of unity, there is some such that . By letting , we see that , which shows that . Then from the definition of we see that for all . However, the are injective and , so . (By the way, this is exactly the same proof that is in Conlon’s book Differentiable Manifolds, just with the other group elements added in. If , then you can take this embedding and project all the points onto the subspace spanned by and you will get his embedding.)
Now we want to show that there is an orthogonal action of on that makes the embedding -equivariant. This is intuitively obvious since we put the in the formula specifically for this purpose, but at first glance it seems like the action of on will be a right action, so I’ll be a bit pedantic just to be careful. (Tim Goldberg helped me out with this part. Until just now I’ve always been confused by the fact that acting on coordinates and acting on basis vectors are “dual” in the same sense that left and right actions are “dual”, and also in the same sense that acting with and are “dual”.) Since we’ve fixed an ordering of our group elements, given any we know that for some permutation . Then we can define a map by , and the question is whether this is a homomorphism or an anti-homomorphism. Intuitively, we’ve “dualized” twice, once by multiplying on the right and the other by inverting, so it should be a homomorphism. Formally,
which shows that . This is the same as , so is a homomorphism. Now acts (on the left) on in the obvious way, by and . This gives a left action of on , i.e. a homomorphism .
Now let’s try to obtain the formula . To simplify the formulas, I’ll just show this for the second term in the formula (which I’ll write as ), which will make it obvious for the first term. We have
(In the fourth equality we’ve used the fact that if we apply to all the index positions where goes, then the sum is unchanged since we are summing over all elements of .) I should mention that this embedding doesn’t help with any kind of visualization since if the group and manifold are both nontrivial then . It does give a nice perspective on compact manifolds with finite group actions, and maybe it helps explain some of the nice properties they enjoy. I don’t know of any uses of this, but I’d be interested in hearing of one if you find one.