Heegaard Floer invariants in codimension one

TL;DR

This paper introduces a new numerical invariant for certain 4-manifolds using Heegaard Floer homology, which helps obstruct specific 3-manifold embeddings and distinguishes exotic smooth structures.

Contribution

It constructs a diffeomorphism invariant for 4-manifolds with the homology of S^1×S^3 using twisted Heegaard Floer d invariants, enabling new embedding obstructions.

Findings

Invariant obstructs embeddings of certain 3-manifolds.

Invariant distinguishes exotic R^4s.

Provides tools for embedding problems in 4-manifold topology.

Abstract

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