Mfcttrf

Write Mod(g,n) for the mapping class group of a genus-$g$ surface $Sigma$ with $n$ boundary components. When $n=0,1$ we define the Torelli group $T$ to be the subgroup of Mod(g,n) which acts trivially on the homology $H = H_1(Sigma,mathbfZ)$.

The Johnson homomorphism is a much-studied homomorphism from the Torelli group to $mathrmHom(H,wedge^2 H)$ (when n=1) or a quotient of this (when n=0) whose kernel turns out to be the commutator subgroup of Torelli.

Morita showed in 1993 that the Johnson homomorphism extends to the whole group Mod(g,1), not as a homomorphism, but as a 1-cocycle in

$H^1(mathrmMod(g,1), mathrmHom(H,wedge^2 H))$

where the action is given by the action of Mod(g,n) on $H$. (Thus the Morita cocycle restricts to a homomorphism on Torelli, as claimed.) It can be thought of as keeping track of the action of the mapping class on the quotient of $pi_1(Sigma)$ by the third term of its lower central series.

All of the above is well-known, or at least well-known to the people who know this kind of thing well. Now here's my question: is there a Morita cocycle on Mod(g,n) when n > 1?

Of course, such a cocycle would restrict to a Johnson homomorphism from the Torelli subgroup of Mod(g,n), and even this is subtle; but Church's paper "Orbits of curves under the Johnson kernel," gives a way to define a Torelli group and a Johnson homomorphism for Mod(g,n) which behaves well with respect to inclusion of subsurfaces. So a more specific version of my question would be: when n > 1, does Church's "Johnson homomorphism" extend to a "Morita cocycle" on all of Mod(g,n) which behaves well with respect to inclusion of subsurfaces?

The answer is "yes" -- in fact one can do better and get a class in

$$H^1(textAut(F_m), textHom(H, wedge^2 H)),$$

where $F_m$ is the free group on $m$ generators and $H$ is the Abelianization of $F_m$, if I'm not mistaken. This gives the cocycle you want since there is an obvious map $textMod_g, nto textAut(pi_1(Sigma_g, n-1))simeq F_2g+n-2$ for $ngeq 2$, given by the conjugation action of $textMod_g,n$ on the point-pushing subgroup, namely $pi_1(Sigma_g, n-1)$.

A construction goes as follows. Let $mathbbZ[F_m]$ be the group ring of $F_m$, and let $mathscrI$ be the augmentation ideal. Then $H simeq mathscrI/mathscrI^2$ canonically (via the map sending $g$ to $g-1$) and $mathscrI^2/mathscrI^3simeq H^otimes 2$ canonically (via the multiplication map). There is a short exact sequence of $textAut(F_m)$ modules $$0to mathscrI^2/mathscrI^3to mathscrI/mathscrI^3to mathscrI/mathscrI^2to 0,$$ which we can think of as an extension of $H$ by $H^otimes 2$, and hence gives a class in

$$H^1(textAut(F_m), textHom(H, H^otimes 2)).$$

But in fact a direct computation of a crossed homomorphism representing this class shows that it lands in $textHom(H, textAlt^2(H)).$