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

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