Poincaré duality algebras mod two

  • Smith L
  • Stong R
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Abstract

We study Poincaré duality algebras over the field F2 of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds. © 2010 Elsevier Inc.

Author-supplied keywords

  • Commutative rings and algebras
  • Polynomial ideals
  • Steenrod algebra

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Authors

  • Larry Smith

  • R. E. Stong

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