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Poincaré duality provides an isomorphism between the homology and cohomology of a compact
manifold, up to a shift. For \pi-finite spaces, i.e. spaces with finitely many non-zero homotopy
groups, all of which are finite, there is a similar duality only for Q-coefficients, 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 \pi-finite spaces with coefficients in
some extra-ordinary cohomology theories called Morava K-theories. This property of Morava
K-theory is called ambidexterity.
I will explain what is ambidexterity and some of its consequences.
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