Topological Membrane Solitons and Loop Orders of Membrane Scatterings in   M-theory
  Two topological issues on membranes in M-theory are studied:   (1) Soliton is an important subject in M-theory. Under the framework of obstruction theory with the help from framed links in $S^3$, we give a complete enumeration of topological membrane solitons in a string-admissible target-space of the form a product of Minkowskian space-times, tori, and K3-surfaces. Patching of these solitons and their topological charges are also defined and discussed.   (2) Loop order of membrane scatterings is the basis for a perturbative M-theory. We explore this concept with emphases on its distinct features from pointlike and stringlike particles.   For completeness, a light exposition on homologies of compact oriented 3-manifolds is given in the Appendix. 
Chien-Hao Liu