Poincaré duality algebras mod two
A commutative graded connected algebra \(H\) over a field \(F\) is said to be a Poincaré duality algebra of formal dimension \(d\) if the following conditions on its homogeneous components are satisfied: (1) \(H_i=0\) for all \(i>d\); (2) \(H_d\) is a one dimensional vector space over \(F\); (3) for every nonzero \(u\in H_i\) there is an element \(\check{u}\in H_{d-i}\), called a Poincaré dual for \(u\), such that \(u\cdot \check{u}\neq 0\) in \(H_d\). In the paper under review Poincaré duality algebras are studied over the field \(\mathbb{F}_2\) of two elements. A Poincaré duality algebra of formal dimension two (three, respectively) is called a surface algebra (a 3-fold, respectively). A complete classification of surface algebras is obtained. At first, a connected sum operation is introduced for Poincaré duality algebras. Then the corresponding Grothendieck group is computed for surface algebras. It is shown that the corresponding group for 3-folds is not finitely generated but does have a Krull-Schmidt property. Isomorphism classes of 3-folds with at most three generators of degree 3 are classified. At the end of the paper several different means are considered of constructing connected sum indecomposable 3-folds.
scientific article; zbMATH DE number 5796753
scientific article