# Adam Levine, Duke, Heegaard Floer invariants for homology $S^1 \times S^3$s

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September 13, 2017 | 4:30 pm - 5:30 pm EDT

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 R^4s. This is joint work with Danny Ruberman.