2009 Topology Seminar - 16

Conjecture: For any map from a closed 4-manifold E to a circle whose homotopy fiber has the homotopy type of a 3-manifold, there exists a fiber bundle over the circle with total space a 4-manifold homotopy equivalent to E. Theorem (joint with Shmuel Weinberger) The conjecture is true when the 3-manifold is a lens space with odd order fundamental group.

The proof involves a surgery theoretic argument which involves a lemma of Gauss used in his third proof of the law of quadratic reciprocity.

This theorem answers a question of Jonathan Hillman, asked in the context of 4-dimensional geometries:

Theorem: Any 4-manifold with Euler characteristic zero and fundamental group a semidirect product where Z acts on Z/odd is homotopy equivalent to a self isometry of a lens space.