Let $K\_g \le \mbox{Mod}\_g$ denote the Johnson kernel subgroup of the mapping class group of a closed surface of genus $g$. [Dimca-Hain-Papadima](https://arxiv.org/pdf/1101.1392.pdf) find an explicit presentation for $H\_1(K\_g; \mathbb{C})$ as a $\mbox{Mod}\_g/K\_g$-module. According to Theorem B of their paper, $H\_1(K\_g;\mathbb{C})$ contains a trivial summand. Passing to $H^1$, this implies that there is some homomorphism $f: K\_g \to \mathbb{C}$ that is *invariant* under the conjugation action of $\mbox{Mod}\_g$ on $K\_g$. 



> 
> Does anyone know an explicit description of this homomorphism?
> 
> 
> 


It is *not* the Casson invariant (which Morita shows determines a (non-canonical) homomorphism $\lambda: K\_g \to \mathbb{Z}$). In a remark in his first paper on the Casson invariant, Morita writes down a formula describing how $\lambda$ is affected by conjugation, and this action is nontrivial (this is ultimately a consequence of the fact that the definition of $\lambda$ requires a non-canonical choice of Heegaard splitting on $S^3$).



A question about the abelianization of the Johnson kernel
I have found an explicit construction in terms of the (higher) Johnson homomorphism. Briefly put, the third Johnson homomorphism is an $\mbox{Sp}\_{2g}(\mathbb{Z})$-equivariant map
$$
\tau\_3: K\_g \to M
$$
for some $\mbox{Sp}\_{2g}(\mathbb{Z})$-module $M$ which can be shown to be isomorphic to $\mbox{Sym}^2(V(2))$ (here we adopt the standard notation for representations of $\mbox{Sp}$). There is a contraction mapping $\mbox{Sym}^2(V(2)) \to \mathbb{C}$, furnishing the required invariant homomorphism.


In case anyone is interested, I will sketch the construction. In the setting where the surface has a boundary component, there's more known about the image of the third Johnson homomorphism. Indeed, Morita shows that 
$$
\tau\_3:H\_1(K\_{g,1}; \mathbb{Q}) \to \mbox{Sym}^2(\wedge^2H)
$$
is a surjection. Here $H:= H\_1(\Sigma\_{g,1};\mathbb{Q})$. We'll work for now over $\mathbb{Q}$. In light of the decomposition $\wedge^2 H = V(2) \oplus V(0)$ into irreps of $\mbox{Sp}\_{2g}(\mathbb{Z})$, the target $\mbox{Sym}^2(\wedge^2 H)$ decomposes as
$$
\mbox{Sym}^2(\wedge^2 H) = \mbox{Sym}^2(V(2)) \oplus V(2) \oplus V(0).
$$
Explicitly, $V(0)$ is spanned by $\omega^2$, where $\omega \in \wedge^2 H$ is the symplectic form. 


There is a short exact sequence
$$
1 \to N \to K\_{g,1} \to K\_g \to 1,
$$
and hence the Johnson homomorphism on $K\_g$ is valued in $\mbox{Sym}^2(\wedge^2 H) / (\tau\_3(N))$. The goal now is to identify the image of $N$. This will be done in two steps, reflecting the fact that there is a short exact sequence
$$
1 \to \mathbb{Z} \to N \to \pi\_1(\Sigma\_g)^{[2]} \to 1,
$$
where the notation $G^{[k]}$ indicates the $k^{th}$ term in the lower central series of a group $G$, taking $G^{[1]} = G$. The kernel $\mathbb{Z}$ is generated by the twist about the boundary component $T\_\Delta$, and $\pi\_1(\Sigma\_g)^{[2]}$ is viewed as a subgroup of $K\_{g,\*}$ via the Birman exact sequence (here $K\_{g,\*}$ is the Johnson kernel for a surface with one puncture).


In the paper mentioned above, Morita shows that
$$
\tau\_3(T\_\Delta) = \omega^2.
$$


This implies that there is a well-defined homomorphism
$$
\tau\_3: H\_1(K\_{g,\*}; \mathbb{Q}) \to \mbox{Sym}^2(V(2)) \oplus V(2).
$$


The next step is to identify the image of $\pi\_1(\Sigma\_g)^{[2]}$ under $\tau\_3$. In fact, the image is $V(2)$. This is because the restriction of $\tau\_3$ to $\pi\_1(\Sigma\_g)^{[2]}$ factors through $\pi\_1(\Sigma\_g)^{[2]} / \pi\_1(\Sigma\_g)^{[3]}$, and 
$$
\pi\_1(\Sigma\_g)^{[2]} / \pi\_1(\Sigma\_g)^{[3]} \otimes \mathbb{Q} \cong V(2).
$$
In summary, we have identified the (rational) image of the third Johnson homomorphism on the closed Johnson kernel:
$$
\tau\_3: K\_g \to \mbox{Sym}^2(V(2)).
$$
Now we simply observe that there is the contraction $C: \mbox{Sym}^2(V(2)) \to \mathbb{Q}$. Thus $C \circ \tau\_3$ is the required invariant homomorphism.


*Morita, Shigeyuki*, [**Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I**](http://dx.doi.org/10.1016/0040-9383(89)90011-6), Topology 28, No.3, 305-323 (1989). [ZBL0684.57008](https://zbmath.org/?q=an:0684.57008).


