Distinguishing closed 4-manifolds by slicing

TL;DR

The paper addresses distinguishing smooth structures on closed 4-manifolds that share the same integral cohomology ring by using the sliceness of a knot as a differentiator. It develops a construction of the 4-manifolds $B$, $W$, and the key piece $V$ via Luttinger surgeries, then shows the figure-eight knot is not slice in $W$ by analyzing the double $Z=V\cup_\sigma V$, a symplectic manifold with $b_2=2$ that is not diffeomorphic to $S^2\times S^2$. Leveraging Heegaard Floer mixed invariants for $b^+=1$ manifolds and a gluing framework, the authors produce exotic smooth structures: a nonstandard cohomology $CP^2\#\overline{CP}^2$ with nonvanishing invariants and an exotic $CP^2\#_5\overline{CP}^2$, including a simple construction of a manifold homeomorphic-but-not-diffeomorphic to $CP^2\#_5\overline{CP}^2$. The results showcase a powerful blend of Luttinger surgery, symplectic topology, and Floer-theoretic tools to differentiate smooth structures in dimension four and to realize new exotic 4-manifolds with prescribed cohomology. Overall, the work expands the methodology for separating homeomorphic-but-not-diffeomorphic 4-manifolds by slicing and Floer-theoretic obstructions, with implications for understanding the smooth classification in four dimensions.

Abstract

One approach to produce a pair of homeomorphic-but-not-diffeomophic closed 4-manifolds is to find a knot which is smoothly slice in one but not the other. This approach has never been run successfully. We give the first examples of a pair of closed 4-manifolds with the same integer cohomology ring where the diffeomorphism type is distinguished by this approach. Along the way, we produce the first examples of 4-manifolds with nonvanishing Seiberg-Witten invariants and the same integer cohomology as $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$ which are not diffeomorphic to $\mathbb{C}P^2\#\overline{\mathbb{C}P^2}$. We also give a simple new construction of a 4-manifold which is homeomorphic-but-not-diffeomorphic to $\mathbb{C}P^2\#5\overline{\mathbb{C}P^2}$.

Figures (3)

• Figure 1:
• Figure 2: R is a genus 2 surface bundle over a puncured torus. The surgery torus $T_\beta$ is the sub-bundle given by the restriction to the two curves marked in blue, and $T_\alpha$ to the curves in green. The orientation reversing involution $\epsilon$ on $F$ is marked in yellow.
• Figure 3: A self diffeomorphism of $S^3_0(Q)$ which changes the parity of the (homology class of the) meridian.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7