Heegaard Floer invariants in codimension one

Abstract

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4 4 -manifold X X with the homology of S 1 × S 3 S^1 \times S^3 . Specifically, we show that for any smoothly embedded 3 3 -manifold Y Y representing a generator of H 3 ( X ) H_3(X) , a suitable version of the Heegaard Floer d d invariant of Y Y , defined using twisted coefficients, is a diffeomorphism invariant of X X . We show how this invariant can be used to obstruct embeddings of certain types of 3 3 -manifolds, including those obtained as a connected sum of a rational homology 3 3 -sphere and any number of copies of S 1 × S 2 S^1 \times S^2 . We also give similar obstructions to embeddings in certain open 4 4 -manifolds, including exotic R 4 \mathbb {R}^4 s.