On the genus of the complex projective plane

Summary

We construct a crystallization Γ of the complex projective plane ℂP ², whose associated pseudocomplex is composed by five vertices, ten edges, twenty triangles, twenty tetrahedra and eight 5-simplexes. This is proved to be the most “economical” pseudodissection of ℂP ². On the other hand, a simple calculation shows that Γ regularly imbeds into the orientable surface of genus 2, with respect to all twelve cyclic permutations of the colour set Δ₄ = {0, 1, 2, 3, 4}. Moreover, by performing the “connected sum” of two copies of Γ, with respect to the two copies of any vertex, we obtain a genus 4 crystallization of$S^2 \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \times } S^2$ (the twisted S²-bundle over S²).

Hence, calling ℒ(M) theregular genus of the manifoldM, i.e. the minimum genus of a surface into which a crystallization ofM regularly imbeds, we obviously have that ℒ(ℂP ²) ⩽ 2 and that ℒ($S^2 \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{ \times } S^2$) ⩽ 4.