Mathematical Field Notes

Gromoll filtration

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In my latest preprint with Georgios Dimitroglou Rizell, we use the topology of diffeomorphism groups of high-dimensional spheres to produce interesting examples of nontrivial topology in symplectomorphism groups of cotangent bundles. Until we started thinking about this, I didn’t know much about the higher homotopy groups of Diff(S^n) so here is some interesting stuff I learned while we were writing this paper.

The mapping class group \pi_0(Diff(S^n)) is a finite abelian group and it comes with a natural filtration by subgroups called the Gromoll filtration (thanks to Oscar Randall-Williams for telling us what it was called – once something has a name it suddenly becomes easier to find out what is known about it). Let me write this (in probably non-standard notation) as:

\cdots \subset G_k(n)\subset G_{k-1}(n)\subset\cdots\subset G_1(n)\subset G_0(n)=\pi_0(Diff(S^n))

The subgroup G_k(n+k) is the image of \pi_k(Diff(S^{n+k})) under a natural homomorphism. The homomorphism is defined as follows. Think of S^{n+k} as (S^n\times D^k)\cup_{S^n\times S^{k-1}} (D^n\times S^{k-1}). Represent an element of \pi_k(Diff(S^n)) as a map F: D^k\to Diff(S^n) (F(y)=\phi_y: S^n\rightarrow S^n) for which S^{k-1}=\partial D^k is sent to 1\in Diff(S^n). Now consider the map \Phi: S^{n+k}\to S^{n+k} which equals the identity on D^n\times S^{k-1} and equals \Phi(x,y)=(\phi_y(x),y)) on S^n\times D^k. For suitably smooth F, \Phi is a diffeomorphism of S^{n+k} and the map [F]\mapsto [\Phi] is the homomorphism whose image is the subgroup G_k(n+k) in the Gromoll filtration.

We can think of this homomorphism \pi_k(Diff(S^n))\rightarrow \pi_0(Diff(S^{n+k})) as a “suspension” map. To understand visually what this suspension looks like, here (see figure) is a familiar example in the case k=1, n=1. There is an obvious loop in Diff(S^1) (namely the loop of rotations: F(y) is the rotation by 2\pi y radians).

 

A (trivial) element of G_1(2)

How elements of G_1 arise from loops in diffeomorphism groups.

 

Suspending this loop by the above prescription gives us the Dehn twist on S^2. Unfortunately the Dehn twist on S^2 is trivial in the mapping class group, but this should give you a sense of how the diffeomorphisms in G_1(n) arise.

In fact, G_1(n) is completely understood. Cerf’s pseudoisotopy theorem implies that G_1(n)=\pi_0(Diff(S^n)): every mapping class of an n-sphere arises by suspending a loop of diffeomorphisms of S^{n-1}, that is “cutting along a S^{n-1} and regluing with a twist”. But to get a nontrivial mapping class, the twist is necessarily more complicated than the loop of rotations.

Nontrivial elements in higher homotopy groups are sometimes fiddly to get one’s hands on; suspending an element \alpha\in\pi_k(Diff(S^{n})) and looking at its image in the mapping class group of an (n+k)-sphere, then using that mapping class as a gluing map to stick two discs together and make an exotic (n+k+1)-sphere lets you turn \alpha into a manifold (S^{n+k+1}_{\alpha} in our notation), from which it is maybe easier to extract an invariant to prove nontriviality. For example, if the resulting exotic sphere does not bound a parallelisable manifold then the original homotopy class must have been nontrivial. This (via an incredible theorem of Abouzaid) is the way the Gromoll filtration enters into our work: if a homotopy class survives in G_k(n+k) modulo the subgroup of G_k(n+k) consisting of exotic spheres bounding parallelisable manifolds then we can use the homotopy class to construct nontrivial topology in the symplectomorphism group of T^*S^n.

So it is important for us to know when G_k(n+k) is nontrivial. Fortunately, there are many interesting calculations of the higher groups G_k in the literature, providing a convenient way to detect nontrivial higher homotopy in the diffeomorphism groups of spheres.

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Written by Jonny Evans

July 14, 2014 at 7:16 am

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