Adam Levine (Princeton University), Triangle Topology Seminar

Title:  Heegaard Floer invariants for homology $S^1 \times S^3$s

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. This is joint work with Danny Ruberman.