Heegaard Floer invariants in codimension one

Abstract

Abstract. Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S 1 × 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 × S 2 . We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R 4 s.

The 0-concordance monoid is infinitely generated

“…The proof of Theorem 1.1 is an easy consequence of previous techniques used by various authors [17], [11], but we have been unable to find a statement of this result in the literature.…”

Invariants of $2$-knots

“…Traditionally, a rational number d(Y, s) ∈ Q is associated to a rational homology 3-sphere Y and a spin-c structure s on Y [105]. This was extended to 3-manifolds with b 1 (Y ) > 0 using twisted coefficients [7,88]. The details are beyond the scope of the survey apart from the fact, due to Levine and Ruberman, that if X is a smooth 4-manifold with the integral homology of S 1 × S 3 , Y ⊂ X is a 3-dimensional submanifold, and s X is the spin-c structure on Y induced from X, then a correction term d(Y, s X ) ∈ Q can be defined and only depends on X and on the homology class y := [Y ] ∈ H 3 (X) [88, Theorem 1.1]; the resulting invariant is denoted d(X, y).…”

“…The details are beyond the scope of the survey apart from the fact, due to Levine and Ruberman, that if X is a smooth 4-manifold with the integral homology of S 1 × S 3 , Y ⊂ X is a 3-dimensional submanifold, and s X is the spin-c structure on Y induced from X, then a correction term d(Y, s X ) ∈ Q can be defined and only depends on X and on the homology class y := [Y ] ∈ H 3 (X) [88, Theorem 1.1]; the resulting invariant is denoted d(X, y). Taking X to be M K , the result of surgery on a smooth 2-knot K ⊂ S 4 , leads to the following definition, which is due to Levine and Ruberman [88]. The d-invariant vanishes for ribbon 2-knots and obstructs invertibility and amphicheirality [88,Section 6]; it also obstructs 0-concordance [126].…”

“…Taking X to be M K , the result of surgery on a smooth 2-knot K ⊂ S 4 , leads to the following definition, which is due to Levine and Ruberman [88]. The d-invariant vanishes for ribbon 2-knots and obstructs invertibility and amphicheirality [88,Section 6]; it also obstructs 0-concordance [126]. Do other homology cobordism invariants of 3-manifolds lead to invariants of 2-knots?…”

A splitting theorem for the Seiberg-Witten invariant of a homology S1× S3

“…It is also known [11,Theorem 3] that it changes sign with the change of orientation and that it is additive with respect to connected sums. A Heegaard Floer version of hpXq was defined in [30] without the assumption that Y be a rational homology sphere.…”