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

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
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