Title: Poincar\'e Duality Algebras Modulo Two and Macaulay's Inverse Systems
Authors: Larry Smith (AG-Invariantentheorie) and R.E. Stong (University of Virginia)
Name of the PDF file: pda\underscore quos.pdf
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If $H$ is a Poincar\'e duality algebra generated by its homogeneous
component of degree $1$ it is called {\bf standardly graded} and the
dimension of its homogeneous component $H_1$ of degree one is called its
{\bf rank}\/. Standardly graded Poincar\'e duality algebras occur as
quotient algebras of a (standardly graded) polynomial algebra by a
maximal primary irreducible ideal Such ideals were studied in the work
of F. S. Macaulay at the start of the last century who developed an
elegant means of constructing them. The fact that these quotients are
Poincar\'e duality algebras is a special case of a result of
W. Gr\"obner.
In this note we study the classification of Poincar\'e duality
algebras over the field $\F_2$ of two elements. We obtain a complete
classification of surfaces, i.e., Poincar\'e duality algebras of formal
dimension two. To do so we determine the Grothendieck group of
standardly graded surface algebras over an arbitrary field under the
operation of connected sum. This group turns out to be $\Z$\/, hence
finitely generated, and mirrors faithfully the topological
classification of closed surfaces. By contrast, for Poincar\'e duality
algebras (standardly graded or not) of formal dimension strictly greater
than two the Grothendieck group fails to be finitely generated.
We make a systematic study of standardly graded threefolds, i.e.,
Poincar\'e duality algebras of formal dimension three that are generated
by their elements of degree one. The isomorphism classes of threefolds
of rank at most three are in bijective correspondence with the orbits of
the action of $\GL(3, \F_2)$ on a $10$-dimensional vector space, the
space of catalecticant matrices. To determine the number of isomorphism
classes we count the number of orbits using invariant theory. As a
byproduct we obtain a classification of arbitrary bilinear forms in up
to three variables.
We determine explicitly all the standardly graded threefolds of rank at
most three. There are 21 isomorphism classes. Twelve of these admit an
unstable Steenrod algebra action, so could in theory be realized as the
mod $2$ cohomology of a closed manifold. We exhibit for each such
example a corresponding manifold; most of these are obvious, but there
is one example of a slightly exotic $3$-manifold that is a torus bundle
over a circle to which we devote some space.
For threefolds of higher rank we explain one of several ways to
construct such algebras that are not connected sums using Macaulay's
theory of inverse systems.