Metric stretching and the period map for smooth 4-manifolds

TL;DR

This paper studies the period map for smooth 4-manifolds, showing its dense image generally and surjectivity in certain cases, using metric stretching techniques to analyze harmonic forms and cohomology.

Contribution

It proves the density of the period map's image for all 4-manifolds and surjectivity when $b^+=1$, extending results to higher dimensions with new metric constructions.

Findings

Period map has dense image for every 4-manifold.

Surjective when $b^+=1$.

Techniques involve stretching metrics along hypersurfaces.

Abstract

The period map for a smooth closed 4-manifold assigns to a Riemannian metric the space of self-dual harmonic 2-forms. This map is from the space of metrics to the Grassmannian of maximal positive subspaces in the second cohomology, where positivity is defined by cup product. We show that the period map has dense image for every 4-manifold, and that it is surjective if $b^+=1$. Similar results hold for manifolds of dimension a multiple of four. The proofs involve families of metrics constructed by stretching along various hypersurfaces.