Let
h
2
{h_2}
be the degree two Siegel space and
S
p
(
4
,
Z
)
Sp(4,\mathbb {Z})
the symplectic group. The quotient
S
p
(
4
,
Z
)
∖
h
2
Sp(4,\mathbb {Z})\backslash {h_2}
can be interpreted as the moduli space of stable Riemann surfaces of genus
2
2
. This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus
2
2
. The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of
S
p
(
4
,
Z
)
Sp(4,\mathbb {Z})
to the cohomology of the genus two mapping class group
Γ
2
0
\Gamma _2^0
. Using this tool, the
3
3
- and
5
5
-primary pieces of the integral cohomology of
S
p
(
4
,
Z
)
Sp(4,\mathbb {Z})
are computed.