The mapping class group can be embedded in the Automorphism group of a free group by classical work of Dehn and Nielsen. Methods used to investigate the mapping class group have frequently and often successfully been adapted into methods used to investigate . The Johnson homomorphism, introduced by Dennis Johnson, provides an example of this bridge. That said – I will define the Johnson homomorphism first for and then recover Johnson’s original definition by restricting attention to the mapping class group.
Automorphism Groups of Free Groups
The abelianisation map induces a homomorphism,
The Torelli group is defined to be the kernel of this homomorphism,
is the subgroup of that acts trivially on the abelianisation . We generalise this definition as follows. Consider the lower central series for , denoted ,
where and . With this notation was defined as the kernel of the map,
Set and define to be the kernel of the homomorphism,
The groups define a descending filtration of ,
called the Andreadakis-Johnson filtration of . One is interested in understanding the graded quotients of this filtration,
as they are in some sense an approximation to . We define the graded quotients and of the lower central series of and of , in the same way as above.
The terms of the Andreadakis-Johnson filtration consist of automorphisms that act trivially on . We use this very fact to define the Johnson homomorphism. Indeed, given an element consider as an element of (that is we are considering the coset ). But and so acts trivially on this quotient, and thus,
That is, is an element of . Therefore, given an element one can define a homomorphism,
The details of showing this is a homomorphism are not so interesting (they can be found for example in ). More interesting however is the fact that we have a homomorphism,
Moreover it is not hard to see that the kernel of this homomorphism is precisely . Indeed just check that if and only if .
We therefore have an injective homomorphism from the quotient,
is the k-th Johnson homomorphism of .
Mapping Class Groups
The mapping class group can be embedded in the automorphism group of the free group as follows. We consider surfaces of genus with one boundary component, . Denote the mapping class group of isotopy classes of orientation preserving homeomorphisms which fix the boundary pointwise (see  for detailed definitions and a beautiful introduction to the subject). An element of the mapping class group therefore induces an action on the fundamental group . We have the following classical result.
Theorem (Dehn-Nielsen) The homomorphism descibrd above is injective for all
Composition with the map (induced from the abelianisation map ) gives a homomorphism,
It turns out that . Unpacking the definitions gives a bit of insight into how this was discovered to be the image. The map is induced from the action of the mapping class group on , and the map is induced from abelianisation. The abelianisation of is , and so we are really just thinking about the action of the mapping class group on . It turns out that is a symplectic vector space the symplectic form coming from algebraic intersection number (again see details in ).
We can also define the Torelli group for the mapping class group as , which is just elements of the mapping class group which induce a trivial action on the homology, . We can go further and define the Johnson filtration of the mapping class group, by setting . We get the descending filtration,
We play a similar game to above and consider the graded quotients of this filtration. Again these are seen as an approximation to the Torelli group of the mapping class group .
- The Johnson homomorphism is injective.
However it is not surjective. It is an open question to determine how far from surjective it is, that is, to understand the image, or equivalently, the cokernel.
Morita defined a map , where denotes symmetric product. is called Morita’s trace map. Morita proved,
- is surjective
This result provides an obstruction to the surjectivity of the Johnson homomorphism (indeed the composition of two surjective maps is surjective), and as such is sometimes called the Morita obstruction.
In the future I hope to elaborate on Morita’s trace map (i.e. actually define it!), and also talk about some more recent trace maps, in particular the trace maps of Enomoto-Satoh and of Conant-Kassabov-Vogtmann. These trace maps provide more insight into the cokernel of the Johnson homomorphism.
1. http://arxiv.org/abs/1204.0876 A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics – Takao Satoh
2. A Primer on Mapping Class Groups – Benson Farb and Dan Margalit