# Mathematical Field Notes

## Gromoll filtration

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).

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.