×

Immersed spheres in 4-manifolds and the immersed Thom conjecture. (English) Zbl 0869.57016

The classical Thom conjecture states that the genus of a smoothly embedded surface \(F\) in \(CP^2\) representing \(d\) times the generator \(H\) of \(H_2 (CP^2;Z)\) must satisfy \(g(F) \geq(d-1) (d-2)/2\). Representing 2-dimensional homology classes by immersed 2-spheres one arrives at the Immersed Thom Conjecture: Suppose that a 2-sphere \(S\) is immersed in \(CP^2\) with \(p\) positive double points, and suppose that its image represents \(dH \in H_2 (CP^2,Z)\). Then \(p\geq (d-1) (d-2)/2\). The authors give a proof of this conjecture.

MSC:

57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
PDFBibTeX XMLCite