Poincaré duality algebras mod two

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Abstract

We study Poincaré duality algebras over the field F2of 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.

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Smith, L., & Stong, R. E. (2010). Poincaré duality algebras mod two. Advances in Mathematics, 225(4), 1929–1985. https://doi.org/10.1016/j.aim.2010.04.013

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