Bounds on genus and geometric intersections from cylindrical end moduli spaces. (English) Zbl 1056.57022

Strle, Sašo

J. Differ. Geom. 65, No. 3, 469-511 (2004).

The author uses Seiberg-Witten theory in order to estimate from below the genus of smooth representatives of 2-dimensional homology classes in 4-dimensional manifolds. Below \(X\) is a smooth closed connected oriented manifold with \(b_1(X)=0\), and the term “characteristic vector” means any class \(c\in H^2(X)\) whose modulo 2 reduction is the Stiefel-Whitney class \(w_2\).
Theorem A. Suppose that \(b_2^+(X)=1\). If \(\Sigma\subset X\) is a smooth embedded surface of positive self-intersection, then \(\chi(\Sigma)+[\Sigma]^2\leq | \langle c, [\Sigma]\rangle| \) for any characteristic vector \(c\) with \(c^2>\sigma(X)\).
Theorem B. Suppose that \(b_2^+(X)=n>1\). Let \(\Sigma_i, i=1, \dots, n\) be disjoint smooth embedded surfaces in \(X\) with positive self-intersections. If \(c\) is a characteristic vector satisfying \(c^2>\sigma(X)\) and \(\langle c, [\Sigma_i]\rangle>0\) for all \(i\), then the inequality \(\chi(\Sigma_i)+[\Sigma_i]^2\leq \langle c, [\Sigma_i]\rangle\) holds for at least one \(i\).

Reviewer: Yuli Rudyak (Gainesville)

MSC:

57R95	Realizing cycles by submanifolds
32J15	Compact complex surfaces
55N35	Other homology theories in algebraic topology
57M99	General low-dimensional topology
57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R40	Embeddings in differential topology
57R57	Applications of global analysis to structures on manifolds

Keywords:

Seiberg-Witten; genus; Thom conjecture