L-S Category Of Toric Manifolds And Construction Of Minimal Covering Of Closed Surfaces

In this paper,we first introduce the brief history of the development of band group acting manifolds,the background of the development of L-S category and the related concepts expressed by its fundamental inequality.Secondly,Secondly,We discuss the L-S category（minimal open coverage）of toric manifolds and quasi-torus manifolds M²ⁿ,and the conclusion is proved:L-S（Mn·）=L-S(M²ⁿ)=n,（n≠0）.Thirdly,we give the classification of two-dimensional closed surfaces,and we get the conclusion for any closed surface nT2 and mP2:L-S（nT2）=L-S（mP2）=3,（n,m≥1）and the conclusion is elaborated and proved in detail.Finally,we geometrically construct the minimal open covering of closed any surfaces.