Knot Floer homology and relative adjunction inequalities (Q2102704)

Let \(X\) be a smooth, closed, oriented \(4\)-manifold. The well-known ``adjunction inequalities'' constrain the genera of smoothly embedded surfaces in \(X\) possessing non-vanishing gauge-theoretic invariants. Namely, if \(X\) satisfies \(b_2^+(X) > 1\), where \(b_2^+(X)\) is the number of positive eigenvalues of the intersection form on \(H_2(X; \mathbb R)\) and if \(\mathfrak t\) is a \(\mathrm{Spin}^c\) structure on \(X\) with non-zero Seiberg-Witten or Ozsváth-Szabó invariant, then \[ |\left\langle c_1(\mathfrak t), [\Sigma]\right\rangle| + [\Sigma] \cdot [\Sigma] \leq 2 g(\Sigma)-2, \] where \(\Sigma \subset X\) is any smoothly embedded oriented surface with non-negative self-intersection. The main result of the paper under review is a relative version of the adjunction inequality for properly embedded surfaces in \(4\)-manifolds with boundary. More precisely, let \(X\) be a smooth, compact, oriented \(4\)-manifold with \(\partial X = -Y_1 \sqcup Y_2\). Let \(K_1 \subset Y_1\) and \(K_2 \subset Y_2\) be rationally null-homologous knots. If \(F_{X,\mathfrak t}(\alpha) = \beta \neq 0\) then \[ |\left\langle c_1(\mathfrak t), [\Sigma]\right\rangle| + [\Sigma] \cdot [\Sigma] + 2(\tau_\beta(Y_2,K_2) - \tau_\alpha(Y_1,K_1)) \leq 2 g(\Sigma), \] where \(\Sigma\) is any oriented surface, smoothly and properly embedded with \(\partial \Sigma = -K_1 \sqcup K_2\). Here \(\tau_\beta(Y,K)\) is a number associated to a non-zero class \(\beta\) in Heegard Floer homology that records the level of \(\beta\) in a filtration on Heegard Floer homology determined by \(K\). The map \(F_{X, \mathfrak t}\) is a cobordism map in Heegard Floer homology induced by \(X\). Applications include producing analogues of the Ozsváth-Szabó-Rasmussen concordance invariant for links. This allows the authors of the paper under review to reprove the link version of the Milnor conjecture and to show that knot Floer homology detects strongly quasipositive fibered links.