# The Johnson homomorphism

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 $Aut(F_n)$. The Johnson homomorphism, introduced by Dennis Johnson, provides an example of this bridge. That said – I will define the Johnson homomorphism first for $Aut(F_n)$ and then recover Johnson’s original definition by restricting attention to the mapping class group.

### Automorphism Groups of Free Groups

The abelianisation map $F_n \rightarrow F_n/F_n' \cong \mathbb{Z}^n$ induces a homomorphism,

$\rho: Aut(F_n) \rightarrow Aut(\mathbb{Z}^n) \cong GL(n, \mathbb{Z})$

The Torelli group is defined to be the kernel of this homomorphism,

$IA_n:= \mbox{ker}(\rho)$

$IA_n$ is the subgroup of $Aut(F_n)$ that acts trivially on the abelianisation $F_n/F_n'$. We generalise this definition as follows. Consider the lower central series for $F_n$, denoted $\{\mathcal{L}_n\}$,

$F_n = \Gamma_n(1) > \Gamma_n(2) > \cdots > \Gamma_n(k) > \cdots$

where $\Gamma_n(2) = [\Gamma_n(1), \Gamma_n(1)]$ and $\Gamma_n(i+1)=[\Gamma_n(1), \Gamma_n(i)]$. With this notation $IA_n$ was defined as the kernel of the map,

$Aut(F_n) \rightarrow Aut(F_n/\Gamma_n(2))$

Set $\mathcal{A}_n(1) = IA_n$ and define $\mathcal{A}_n(k)$ to be the kernel of the homomorphism,

$Aut(F_n) \rightarrow Aut(F_n/\Gamma_n(k+1))$

The groups $\mathcal{A}_n(k)$ define a descending filtration of $IA_n$,

$\mathcal{A}_n(1)\supseteq\mathcal{A}_n(2)\supseteq\cdots\supseteq\mathcal{A}_n(k)\supseteq\cdots$

called the Andreadakis-Johnson filtration of $Aut(F_n)$. One is interested in understanding the graded quotients of this filtration,

$gr^k(\mathcal{A}_n):=\mathcal{A}_n(k)/\mathcal{A}_n(k+1)$

as they are in some sense an approximation to $IA_n$. We define the graded quotients $gr^k(\mathcal{L}_n)$ and $gr^k(\mathcal{L}_{IA_n})$ of the lower central series of $Aut(F_n)$ and of $IA_n$, in the same way as above.

The terms of the Andreadakis-Johnson filtration $\mathcal{A}_n(k)$ consist of automorphisms $\phi \in Aut(F_n)$ that act trivially on $F_n/\Gamma_n(k+1)$. We use this very fact to define the Johnson homomorphism. Indeed, given an element $z(\mbox{mod}~\Gamma_n(2)) \in F_n/F'_n = gr^1(\mathcal{L}_n)$ consider $z^{-1}\phi(z)$ as an element of $F_n/\Gamma_n(k+1)$ (that is we are considering the coset $z^{-1}\phi(z)(\mbox{mod}~\Gamma_n(k+1))$). But $\phi \in \mathcal{A}_n(k)$ and so acts trivially on this quotient, and thus,

$z^{-1}\phi(z)(\mbox{mod}~\Gamma_n(k+1))=z^{-1}z(\mbox{mod}~\Gamma_n(k+1))=1(\mbox{mod}~\Gamma_n(k+1))$

That is, $z^{-1}\phi(z)$ is an element of $\Gamma_n(k+1)$. Therefore, given an element $\phi \in \mathcal{A}_n(k)$ one can define a homomorphism,

$\widetilde{\tau_k}(\phi): gr^1(\mathcal{L}_n)\rightarrow gr^{k+1}(\mathcal{L}_n)$

$z(\mbox{mod}~\Gamma_n(2)) \mapsto z^{-1}\phi(z)(\mbox{mod}~\Gamma_n(k+2))$

The details of showing this is a homomorphism are not so interesting (they can be found for example in [1]). More interesting however is the fact that we have a homomorphism,

$\widetilde{\tau_k}:\mathcal{A}_n(k)\rightarrow\mbox{Hom}_{\mathbb{Z}}(gr^1(\mathcal{L}_n), gr^{k+1}(\mathcal{L}_n))$

$\phi \mapsto \widetilde{\tau_k}(\phi)$

Moreover it is not hard to see that the kernel of this homomorphism is precisely $\mathcal{A}_n(k+1)$. Indeed just check that $z^{-1}\phi(z) \in \Gamma_n(k+2)$ if and only if $\phi \in \mathcal{A}_n(k+1)$.

We therefore have an injective homomorphism from the quotient,

$\tau_k: gr^k(\mathcal{A}_n)\hookrightarrow Hom_{\mathbb{Z}}(gr^1(\mathcal{L}_n),gr^{k+1}(\mathcal{L}_n))$

$\tau_k$ is the k-th Johnson homomorphism of $Aut(F_n)$.

### 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 $g$ with one boundary component, $S_{g,1}$. Denote the mapping class group $Mod(g,1)$ of isotopy classes of orientation preserving homeomorphisms $S_{g,1}\rightarrow S_{g,1}$ which fix the boundary pointwise (see [2] 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 $\pi_1(S_{g,1})\cong F_{2g}$. We have the following classical result.

Theorem (Dehn-Nielsen)  The homomorphism $\psi:Mod(g,1)\rightarrow Aut(F_{2g})$ descibrd above is injective for all $g\geq1$

Composition with the map $\rho:Aut(F_{2g}) \rightarrow GL(2g, \mathbb{Z})$ (induced from the abelianisation map $F_{2g}\rightarrow \mathbb{Z}^{2g}$) gives a homomorphism,

$\Psi=\rho\cdot\phi: Mod(g,1) \rightarrow GL(2g,\mathbb{Z})$

It turns out that $Im(\Psi)=Sp(2g, \mathbb{Z})$. Unpacking the definitions gives a bit of insight into how this was discovered to be the image. The map $\psi$ is induced from the action of the mapping class group on $\pi_1(S_{g,1})$, and the map $\rho$ is induced from abelianisation. The abelianisation of $\pi_1(S_{g,1})$ is $H:=H_1(S_{g,1};\mathbb{Z})$, and so we are really just thinking about the action of the mapping class group on $H$. It turns out that $H$ is a symplectic vector space the symplectic form coming from algebraic intersection number (again see details in [2]).

We can also define the Torelli group for the mapping class group as $\mathcal{I}_{g,1}:= Mod(g,1)\cap IA_{2n}$, which is just elements of the mapping class group which induce a trivial action on the homology, $H$. We can go further and define the Johnson filtration of the mapping class group, by setting $\mathcal{M}_{g,1}(k)=Mod(g,1)\cap\mathcal{A}_n(k)$. We get the descending filtration,

$\mathcal{I}_{g,1}=\mathcal{M}_{g,1}(1)\supseteq\mathcal{M}_{g,1}(2)\supseteq\mathcal{M}_{g,1}(3)\supseteq\cdots$

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 $\mathcal{I}_{g,1}$.

### Some Results

• The Johnson homomorphism $\tau_k: gr^k(\mathcal{A}_n)\hookrightarrow Hom_{\mathbb{Z}}(H, gr^{k+1}(\mathcal{L}_n))$ 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 $Tr_k:Hom(H, gr^{k+1}(\mathcal{L}_n))\rightarrow S^kH$, where $S^kH$ denotes symmetric product. $Tr_k$ is called Morita’s trace map. Morita proved,

• $Tr_k$ is surjective
• $Tr_k\circ\tau_k\equiv0$

This result provides an obstruction to the surjectivity of the Johnson homomorphism (indeed the composition of two surjective maps is surjective), and as such $S^kH$ 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.

### References

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