I don't know what $H_3(Sp_{2g}(Z))$ is, but I want to give a reference for a related fact. Let $Mod_g$ be the mapping class group. The action on $H_1$ of the surface preserves the algebraic intersection form and gives a representation $Mod_g\to Sp_{2g}(Z)$ which is well-known to be surjective. Rationally, the stable cohomology of $Sp_{2g}(Z)$ injects into the stable cohomology of $Mod_{g}$ and gives "one half" of the stable cohomology of $Mod_{g}$.

Anyway, I do know a reference for some stable integral homology calculations for the mapping class group. Namely, in The low-dimensional homotopy of the stable mapping class group, Ebert proves that in a stable range we have $H_3$ of the mapping class group equal to $Z/12Z$ and $H_4$ equal to $Z^2$. He does this by using the fact that Madsen and Weiss's work identifies the infinite loop space $(Mod_{\infty})^+$ with a space whose first few homotopy groups are known. Ebert then builds with his bare hands the first few stages of the Postnikov tower for this space.

Presumably this either has been done or could be done for $Sp_{2g}(Z)$ by a sufficiently motivated homotopy theorist.

I don't know what $H_3(Sp_{2g}(Z))$ is, but I want to give a reference for a related fact. Let $Mod_g$ be the mapping class group. The action on $H_1$ of the surface preserves the algebraic intersection form and gives a representation $Mod_g\to Sp_{2g}(Z)$ which is well-known to be surjective. Rationally, the stable cohomology of $Sp_{2g}(Z)$ injects into the stable cohomology of $Mod_{g}$ and gives "one half" of the stable cohomology of $Mod_{g}$.

Anyway, I do know a reference for some stable integral homology calculations for the mapping class group. Namely, in The low-dimensional homotopy of the stable mapping class group, Ebert proves that in a stable range we have $H_3$ of the mapping class group equal to $Z/12Z$ and $H_4$ equal to $Z^2$. He does this by using the fact that Madsen and Weiss's work identifies the infinite loop space $(Mod_{\infty})^+$ with a space whose first few homotopy groups are known. Ebert then builds with his bare hands the first few stages of the Postnikov tower for this space.

Presumably this either has been done or could be done for $Sp_{2g}(Z)$ by a sufficiently motivated homotopy theorist.

I don't know what H_3(Sp_{2g}(Z)) is, but I want to give a reference for a related fact. Let Mod_g be the mapping class group. The action on H_1 of the surface preserves the algebraic intersection form and gives a representation Mod_g-->Sp_{2g}(Z) which is well-known to be surjective. Rationally, the stable cohomology of Sp_{2g}(Z) injects into the stable cohomology of Mod_{g} and gives "one half" of the stable cohomology of Mod_{g}.

Anyway, I do know a reference for some stable integral homology calculations for the mapping class group. Namely, in this paper, Ebert proves that in a stable range we have H_3 of the mapping class group equal to Z/12Z and H_4 equal to Z^2. He does this by using the fact that Madsen and Weiss's work identifies the infinite loop space (Mod_{\infty})^+ with a space whose first few homotopy groups are known. Ebert then builds with his bare hands the first few stages of the Postnikov tower for this space.

Presumably this either has been done or could be done for Sp_{2g}(Z) by a sufficiently motivated homotopy theorist.

I don't know what $H_3(Sp_{2g}(Z))$ is, but I want to give a reference for a related fact. Let $Mod_g$ be the mapping class group. The action on $H_1$ of the surface preserves the algebraic intersection form and gives a representation $Mod_g\to Sp_{2g}(Z)$ which is well-known to be surjective. Rationally, the stable cohomology of $Sp_{2g}(Z)$ injects into the stable cohomology of $Mod_{g}$ and gives "one half" of the stable cohomology of $Mod_{g}$.

Anyway, I do know a reference for some stable integral homology calculations for the mapping class group. Namely, in The low-dimensional homotopy of the stable mapping class group, Ebert proves that in a stable range we have $H_3$ of the mapping class group equal to $Z/12Z$ and $H_4$ equal to $Z^2$. He does this by using the fact that Madsen and Weiss's work identifies the infinite loop space $(Mod_{\infty})^+$ with a space whose first few homotopy groups are known. Ebert then builds with his bare hands the first few stages of the Postnikov tower for this space.

Presumably this either has been done or could be done for $Sp_{2g}(Z)$ by a sufficiently motivated homotopy theorist.