Tag Archives: putman

Homology of the Torelli group and negative-dimensional vector spaces

OK, not really.  You know and I know there’s no such thing as a negative-dimensional vector space.

And yet…

The Torelli group T_g is a subject of hot interest to mapping class groups people — it’s the kernel of the natural surjection from the mapping class group Γ_g to Sp_{2g}(Z).  You can think of it as “the part of the mapping class group that arithmetic lattices can’t see,” or at least can’t see very well, and as such it is somewhat intimidating.  We know very little about it, even in small genera.  One thing we do know is that for g at least 3 the Torelli group is finitely generated; this is a theorem of Johnson, and a recent paper by Andy Putman provides a small generating set.  So H_1(T_g,Q) is finite-dimensional.  (From now on all cohomology groups will be silently assigned rational coefficients.)

But a charming argument of Akita shows that, in general, T_g has some infinite-dimensional homology groups.  How do we know?  Because if it didn’t, you would be able to compute the integer χ(T_g) from the formula

χ(T_g) =  χ( Γ_g)/ χ(Sp_{2g}(Z)).

But both the numerator and denominator of the right-hand-side are known, and their quotient is not an integer once g is at least 7.  Done!

At the Park City Mathematics Institute session I visited this summer, there was a lot of discussion of what these infinite-dimensional homology groups of Torelli might look like.  We should remember that the outer action of Sp_2g(Z) on Torelli yields an action of Sp_2g(Z) on the homology of Torelli — so one should certainly think of these spaces as representations of Sp_2g(Z), not as naked vector spaces.  In the few cases these groups have been described explicitly, they are induced from finite-dimensional representations of infinite-index subgroups H of Sp_2g(Z).

I just wanted to record the small observation that in cases like this, there’s a reasonably good way to assign a “dimension” to the homology group!  Namely:  suppose G is a discrete group and H a a subgroup, and suppose that both BG and BH are homotopic to finite complexes.  (This is not quite true for G = Sp_2g(Z), but surely you’re willing to spot me a little finite level structure wherever I need it.)  Let W be a finite-dimensional representation of H and let V be the induction of W up to G.

Now if H were finite-index in G you’d have

dim V = [G:H] dim W

or, what’s the same,

dim V = χ(BH)/χ(BG) dim W

But note that the latter formula makes sense even if H is infinite-index in G!  And this allows you to assign a “dimension” to some infinite-dimensional homology groups.

For instance, consider T_2, which is not finitely generated.  By a theorem of Mess, it’s a free group on a countable set of generators; these generators are naturally in bijection with cosets in Sp_4(Z) of a subgroup H containing SL_2(Z) x SL_2(Z) with index 2.  Compute the Euler characteristics of H and Sp_4 and you find that the “dimension” of H_1(T_2) is -5.

And when you ask Akita’s argument about this case, you find that the purported Euler characteristic of T_2 is 6; a perfectly good integer, but not such a great Euler characteristic for a free group to have.  Unless, of course, it’s a free group on -5 generators.

If you want to see this stuff written up a bit (but only a bit) more carefully, here’s a short .pdf version, which also includes a discussion of the hyperelliptic Torelli group in genus 3.

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Back from Boston

I spent the end of last week in Cambridge, where I gave a talk at MIT about the homology of Hurwitz spaces.

  • Good addition to Harvard Square: there’s now a used-book table on Mass Ave, like the ones I used to patronize on Broadway in Morningside Heights. I flipped through “Strike From Space,” a 1965 book by Phyllis Schlafly and Lester Ward, with the thesis that dopey Democrats were getting head-faked by the Soviets into escalating our commitment to Vietnam, which conflict was in fact not the firewall against an all-red Asia but a mere Soviet plot to distract us from an imminent nuclear first strike from orbit. Bad addition to Harvard Square: a Qdoba! I thought fast food was forbidden from Harvard Square by zoning laws, but apparently you can apply for an exemption — here’s Chipotle’s application to open a restaurant at One Brattle Square.
  • I had the pleasure of meeting Andy Putman and learning about a beautiful recent theorem of his; the moduli space of genus-g curves with full level N structure has the same second homology as M_g itself, for g at least 5. As a corollary, all these spaces have Picard number 1. I also learned a lot from Denis Auroux about mapping class group factorizations and symplectic Lefschetz fibrations — but more on this when I write a post or two on Hurwitz spaces.  Update: (25 Sep 08) Putman’s paper is now on the arXiv.
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