Poincaré duality provides an isomorphism between the homology and cohomology of a compact
manifold, up to a shift. For \pifinite spaces, i.e. spaces with finitely many nonzero homotopy
groups, all of which are finite, there is a similar duality only for Qcoefficients, but no such
duality exists with F_p coefficients. However, as shown by Michael Hopkins and Jacob Lurie,
there is a duality between the homology and cohomology of \pifinite spaces with coefficients in
some extraordinary cohomology theories called Morava Ktheories. This property of Morava
Ktheory is called ambidexterity.
I will explain what is ambidexterity and some of its consequences.
