Poincaré duality algebras mod two

This paper concerns Poincar´e duality algebras [D. M. Meyer and L. Smith, Poincar´e duality algebras, Macaulay’s dual systems, and Steenrod operations, Cambridge Tracts in Math., 167, Cambridge Univ. Press, Cambridge, 2005; MR2177162 (2006h:13012)] over the Galois field F2. Unless otherwise stated, the algebras are standard graded. Those of formal dimension 2 and 3 are called surface algebras and threefolds, respectively. The authors introduce the operation of connected sum of Poincar´e duality algebras of the same formal dimension. This operation turns the isomorphism classes of Poincar´e duality algebras into a commutative torsion-free monoid. The authors study the Grothendieck group corresponding to the submonoid of the standard graded Poincar´e duality algebras under connected sum. Using the notion of catalecticant matrices associated with a standard graded Poincar´e duality algebra, they show that the Grothendieck group of surface algebras is finitely generated and mirrors faithfully the topological classification of closed surfaces. Then they prove that the Grothendieck group for algebras of formal dimension d > 2 is free abelian, but not finitely generated. For Poincar´e duality quotient algebras H = F2[x, y]/I of formal dimension 2 there are 3 isomorphism classes. The case of threefolds of formal dimension 3 is richer and more complicated and to these algebras is the rest of the paper dedicated. It turns out that there are 21 isomorphism classes of standard graded Poincar´e duality algebras of formal dimension 3 and rank at most 3. The authors use Macaulay’s Double Duality Theorem, which allows them to reduce the problem to several invariant theoretic problems. They describe each of the 21 isomorphism classes. Twelve of these admit an unstable Steenrod algebra action, and so could be realized as the mod 2 cohomology of a closed manifold. They show for each such example a corresponding manifold; some space is devoted to one of these, a 3-manifold that is a torus bundle over a circle. They also give constructions for non-standard threefolds indecomposable with respect to the connected sum. The closing section and Appendix A provide several different means of constructing connected sum indecomposable threefolds.