Minimal genus in \(S^1\times M^3\) (Q1283509)

An important problem in 4-dimensional topology is to determine the minimal genus of an embedded surface which represents a given homology class. There have been substantial recent advances on this problem using the techniques of gauge theory, with many of the main results coming through joint work of the author and T. Mrowka. A powerful tool here is the adjunction inequality \(\chi(\Sigma) \geq \sigma \cdot \sigma+\tau \cdot \sigma\); here \(\tau\) represents a Seiberg-Witten basic class for a spin\(^c\) structure, \(\chi(\Sigma)=2g(\Sigma)-2\) for surfaces \(\Sigma\) of positive genus and square, and \(\sigma\) denotes the homology class of the surface \(\Sigma\). A similar formula was first proved by the author and \textit{T. S. Mrowka} [J. Differ. Geom. 41, No. 3, 573-734 (1995; Zbl 0842.57022)] using Donaldson polynomial invariants; the Seiberg-Witten version is due to the author and \textit{T. S. Mrowka} [Math. Res. Lett. 1, No. 6, 797-808 (1994; Zbl 0851.57023)] and to \textit{J. W. Morgan, Z. Szabo}, and \textit{C. H. Taubes} [A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Differ. Geom. 44, No. 4, 706-788 (1996)]. This result is not as useful when the Seiberg-Witten invariants contain little information, such as manifolds of the form \(S^1 \times M^3\) which are studied here. The main result of this paper is a similar inequality \(\chi_{-}(\Sigma) \geq \sigma \cdot \sigma+\varepsilon \cdot \sigma\) for manifolds \(S^1 \times M^3\), where \(M^3\) is a closed, irreducible, oriented 3-manifold carrying a smooth, taut foliation \({\mathcal F}\) by oriented 2-dimensional leaves and \(\varepsilon \) is the pullback of the Euler class \(e(T{\mathcal F})\) of the foliation. Here \(\chi_{-}(\Sigma)\) is defined for a (not necessarily connected) surface \(\Sigma\) as the sum \(\sum_{g_i > 0} 2g_i-2\), with \(g_i\) denoting the genus of the component \(\Sigma_i\) of the surface. Results of \textit{W. P. Thurston} [Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)] and \textit{D. Gabai} [J. Differ. Geom. 18, 445-503 (1983; Zbl 0533.57013)] imply this result for the special case of classes coming from \(M\). The result is deducible from the adjunction inequality when \(\varepsilon \) is a basic class, but can provide stronger results when it isn't. An interesting example which is discussed is when \(M\) is 0-surgery on a knot. Then work of \textit{G. Meng} and \textit{C. H. Taubes} [Math. Res. Lett. 3, No. 5, 661-674 (1996; Zbl 0870.57018)] shows that the Seiberg-Witten invariant is carried by the Alexander polynomial, and the adjunction inequality gives the result \(\chi_{-}(\Sigma) \geq 2mn+(2r-2)n\) when \(\sigma=m([S^1] \times \delta)+n\tau\). Here \(\delta,\tau\) are generators for the first and second homology, and \(r\) is the degree of the symmetrized Alexander polynomial. The result proved here is \(\chi(\Sigma) \geq 2mn+(2g-2)n\) where \(g\) is the genus of the knot, and it is sharp. The genus of the knot is always greater than or equal to \(r\), but there are examples of knots with trivial Alexander polynomial and arbitrary genus. Note that \(g=r\) for fibered knots, but then \(S^1 \times M\) is symplectic and so the adjunction inequality should give the best result. The equality of \(g\) and \(r\) is shown to provide an necessary condition to the existence of a symplectic structure on \(S^1 \times M\) when \(M\) is zero surgery on a knot. Using results of Taubes \textit{R. Fintushel} and \textit{R. J. Stern} [Invent. Math. 134, No. 2, 363-400 (1998; Zbl 0914.57015)] had shown that the symmetrized Alexander polynomial being monic is also a necessary condition. The proof of the main result is very interesting in that it provides a nice mix of three- and four-dimensional arguments, as well as contact and symplectic geometry. The Thurston norm plays a key role -- the author and \textit{T. S. Mrowka} had previously developed connections between the Thurston norm and scalar curvature [Math. Res. Lett. 4, No. 6, 931-937 (1997; Zbl 0892.57011)]. Using this approach, the author first gives a new proof of the adjunction inequality based on studying \( | \alpha| ^+ = 4\pi\sqrt{2} \sup_h (\| \alpha^+\| _h/\| s_h\|_h)\). Here \(s_h\) is the scalar curvature in the metric \(h\) and \(\|\alpha\|_h \) denotes the norm using the \(L^2\)-norm of a harmonic representative of \(\alpha \in H^2(X^4,R)\) with superscript \(+\) referring to the self-dual part. A class \(\alpha\) is called a monopole class if it arises as \(c_1(\mathbf c)\) if the Seiberg-Witten equations have a solution for all Riemannian metrics \(h\) on \(X.\) It is shown that \(| \alpha| ^+ \leq 1\) when \(\alpha\) is a monopole class, and this is used as a basic step in proving the adjunction inequality. The proof of the main result uses a similar track of establishing an inequality \(| \varepsilon_k^+| \leq 1, \) for all \(k \geq 0\), where the subscript \(k\) refers to forming connected sum with \(\overline{\mathbb{C} P}^2 k\) times. The techniques of the author's paper with \textit{T. S. Mrowka} on monopoles and contact structures [Invent. Math. 130, No. 2, 209-255 (1997; Zbl 0892.53015)] play a central role in the remainder of the proof. The assumed taut foliation gives rise to a contact structure on \(M\) in each of its orientations, and these lead to symplectic structures on cones \(Z_{\pm} = [\pm 1,\pm \infty) \times M.\) These cones then form the ends of a family of symplectic 4-manifolds \(Z^N_k\) with middle coming from cutting open \(N\) fold covers of \(S^1 \times M \# k \overline{\mathbb{C} P}^2\) at \(M.\) These are studied to provide a proof of the result.