## 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 so here is some interesting stuff I learned while we were writing this paper.

The mapping class group 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:

The subgroup is the image of under a natural homomorphism. The homomorphism is defined as follows. Think of as . Represent an element of as a map () for which is sent to . Now consider the map which equals the identity on and equals on . For suitably smooth , is a diffeomorphism of and the map is the homomorphism whose image is the subgroup in the Gromoll filtration.

We can think of this homomorphism as a “suspension” map. To understand visually what this suspension looks like, here (see figure) is a familiar example in the case , . There is an obvious loop in (namely the loop of rotations: is the rotation by radians).

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

In fact, is completely understood. Cerf’s pseudoisotopy theorem implies that : **every mapping class of an -sphere arises by suspending a loop of diffeomorphisms of , that is “cutting along a 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 and looking at its image in the mapping class group of an -sphere, then using that mapping class as a gluing map to stick two discs together and make an exotic -sphere lets you turn into a manifold ( 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 modulo the subgroup of consisting of exotic spheres bounding parallelisable manifolds then we can use the homotopy class to construct nontrivial topology in the symplectomorphism group of .

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

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