Heegaard Floer invariants in codimension one

Levine, Adam; Ruberman, Daniel

doi:10.1090/tran/7345

Heegaard Floer invariants in codimension one
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by Adam Simon Levine and Daniel Ruberman PDF

Trans. Amer. Math. Soc. 371 (2019), 3049-3081 Request permission

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.

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