called the Lefschetz mapof. Letting be the th iteration of , we have for each a map
Ifiscompact and oriented, then Poincaré duality tells us that and are vector spaces of the same dimension, so in these cases it is natural to ask whether or not the various iterations of Lefschetz maps are isomorphisms.
If
and
are isomorphisms, then is a Lefschetz element, or Lefschetz class. If
is an isomorphism for all , then is a strong Lefschetz element, or a strong Lefschetz class.
Let be a -dimensional symplectic manifold. (Symplectic manifolds are always orientable, although certainly not always compact.) Then is a Lefschetz manifoldif is a Lefschetz element, and is a strong Lefschetz manifoldif is a strong Lefschetz element.
Where to find Lefschetz manifolds
The real manifold underlying any Kähler manifold is a symplectic manifold. The strong Lefschetz theorem tells us that it is also a strong Lefschetz manifold, and hence a Lefschetz manifold. Therefore we have the following chain of inclusions.
In[1], Chal Benson and Carolyn S. Gordon proved that if a compactnilmanifold is a Lefschetz manifold, then it is diffeomorphic to a torus. The fact that there are nilmanifolds that are not diffeomorphic to a torus shows that there is some space between Kähler manifolds and symplectic manifolds, but the class of nilmanifolds fails to show any differences between Kähler manifolds, Lefschetz manifolds, and strong Lefschetz manifolds.
Gordan and Benson conjectured that if a compactcomplete solvmanifold admits a Kähler structure, then it is diffeomorphic to a torus. This has been proved. Furthermore, many examples have been found of solvmanifolds that are strong Lefschetz but not Kähler, and solvmanifolds that are Lefschetz but not strong Lefschetz. Such examples can be found in [2].
Notes
^C. Benson and C. Gordon, Kahler and symplectic structures on nilmanifolds, Topology 27 (1988), 513-518.
^Takumi Yamada, Examples of Compact Lefschetz Solvmanifolds, Tokyo J. Math Vol. 25, No. 2, (2002), 261-283.