Shadow-complexity and trisection genus

2024/07/31 by Naoe, Hironobu, Ogawa, Masaki
#57K41 #57Q15 (Primary) 57R65 (Secondary) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2407.21265

Abstract

The shadow-complexity is an invariant of closed 4-manifolds defined by using 2-dimensional polyhedra called Turaev's shadows, which, roughly speaking, measures how complicated a 2-skeleton of the 4-manifold is. In this paper, we define a new version sc_r of shadow-complexity depending on an extra parameter r≥0, and we investigate the relationship between this complexity and the trisection genus g. More explicitly, we prove an inequality g(W) ≤ 2+2sc_r(W) for any closed 4-manifold W and any r≥1/2. Moreover, we determine the exact values of sc_1/2 for infinitely many 4-manifolds, and also we classify all the closed 4-manifolds with sc_1/2≤1/2.

Related

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• The special shadow-complexity of #_k(S¹× S³)
• A short introduction to shadows of 4-manifolds
• Mazur manifolds and corks with small shadow complexities
• Branched shadows and complex structures of 4-manifolds