15.01.25

Alexander Mrinski (KIT)

14:00-15:00

Which finite CW complexes are homotopy equivalent to closed smooth manifolds?

A necessary condition is that such a complex has the homotopy type of a Poincaré complex, i.e. it satisfies an appropriate form of Poincaré duality with local coefficients. In dimension 4k, there is a further important property of oriented closed smooth manifolds: the signature of their intersection form is multiplicative under finite coverings. One might therefore expect the same behavior for oriented Poincaré complexes.

Building on Wall’s 1967 paper Poincaré complexes, we construct 4-dimensional Poincaré complexes for which this multiplicativity fails. As a consequence, we obtain explicit examples of Poincaré complexes that are not homotopy equivalent to any closed smooth manifold.

SR 3.61