A new invariant of 4-manifolds. (English) Zbl 1421.57031

Kirby, Robion; Thompson, Abigail

Proc. Natl. Acad. Sci. USA 115, No. 43, 10857-10860 (2018).

Summary: We define an integer invariant \(L_X\) of a smooth, compact, closed 4-manifold \(X\) by minimizing a certain complexity of a trisection of \(X\) over all trisections. The good feature of \(L_X\) is that when \(L_X=0\) and \(X\) is a homology 4-sphere, then \(X\) is diffeomorphic to the 4-sphere. Naturally, \(L\) is hard to compute.

MSC:

57N13	Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R65	Surgery and handlebodies

Keywords:

trisections; 4-manifolds; Heegaard splittings; curve complex