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low-dim homology of Sp(2n,Z)

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mathreader

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Oct 16, 2009, 9:29:26 AM10/16/09
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Let V be an irreducible Sp(2n,C)-module with highest weight.

It is known that H^1(Sp(2n,Z),V) = 0 for all n>1, and H^2(Sp(2n,Z),V)
= 0 for n>7.

The first result follows from

Raghunathan, M. S.
Cohomology of arithmetic subgroups of algebraic groups. I, II.
Ann. of Math. (2) 86 (1967), 409-424; ibid. (2) 87 1967 279--304.

And the second from

Borel, Armand
Stable real cohomology of arithmetic groups. II. Manifolds and Lie
groups (Notre Dame, Ind., 1980), pp. 21--55,
Progr. Math., 14, Birkh�user, Boston, Mass., 1981.

.
Are similar results known about the *homology* groups: H_1(Sp(2n,Z),V)
and H_2(Sp(2n,Z),V)? (The action on module V is nontrivial so
universal coefficients do not apply.)

mathreader

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Dec 7, 2009, 3:27:19 AM12/7/09
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On Oct 16, 8:29 am, mathreader <mathrea...@gmail.com> wrote:
> Let V be an irreducible Sp(2n,C)-module with highest weight.
>
> It is known that H^1(Sp(2n,Z),V) = 0 for all n>1, and H^2(Sp(2n,Z),V)
> = 0 for n>7.
>
> The first result follows from
>
> Raghunathan, M. S.
> Cohomology of arithmetic subgroups of algebraic groups. I, II.
> Ann. of Math. (2) 86 (1967), 409-424; ibid. (2) 87 1967 279--304.
>
> And the second from
>
> Borel, Armand
> Stable real cohomology of arithmetic groups. II. Manifolds and Lie
> groups (Notre Dame, Ind., 1980), pp. 21--55,
> Progr. Math., 14, BirkhŠuser, Boston, Mass., 1981.

>
> .
> Are similar results known about the *homology* groups: H_1(Sp(2n,Z),V)
> and H_2(Sp(2n,Z),V)? (The action on module V is nontrivial so
> universal coefficients do not apply.)

Answering my own question:

There is an isomorphism: H^i(G,V') --> H_i(G,V)', where V' is the dual
module V'=Hom(V,k) and k is any field.
The proof is in Brown's book "Cohomology of groups", ch.VI,
Proposition 7.1.

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