The Heegaard Floer group HF^oo(Y, s_0) is known to depend only on the triple cup product of Y and c_1(s_0). In particular, it appears to carry purely cohomological information. A first approximation is given by the 'cup homology' HC(Y), the E^4 page of a spectral sequence to HF^oo. It is conjectured that this spectral sequence collapses. Millions of computer examples in the range 9 <= b_1(Y) <= 14 give that in all cases, HF^oo(Y) is isomorphic to HC(Y) as an abelian group --- a stronger result than we should expect just from knowing that the spectral sequence collapses. One might expect there is a good reason these are isomorphic.

I will prove that there is not by showing these functors are not naturally isomorphic over F_2. Along the way, we introduce a 'Lefschetz decomposition' of exterior algebras over Z.

This work is joint with Analisa Faulkner Valiente.