Singularities of 2-spheres in 4-space and cobordism of knots

Consider an oriented 2-dimensional manifold m imbedded as a subcomplex in a triangulated oriented 4-dimensional manifold M in such a way that the boundary of m is contained in the boundary of M and the interior of m is contained in the interior of M. We will assume that M is a "piecewise linear manifold": that is, the star neighborhood of any point should be piecewise linearly homeomorphic to a 4-simplex.One can measure the local singularity of the imbedding at an interior point x of m as follows.Let N denote the star neighborhood of x in M. The boundary S=dN of N is a 3-shere with an orientation inherited from that of M, and k=mdNis a 1-sphere with an orientation inherited from that of m.The oriented knot type K of the imbedding of k in S is called 20 the singularity of the imbedding at x.When k is of trivial type in dN we may say that the singularity is 0 or that x is a non-singular point or that m is locally flat at x.A surface m is called locally flat if it is locally flat at each of its points.REMARK.The singularity of m at x is clearly a combinatorial invariant of M,m,x\ that is it is not altered if we subdivide M rectilinearly.We do not know whether or not this singularity is a topological invariant, except in the special case of a locally flat point.The topological invariance of the concept of local flatness is easily proved, making use of Dehn's lemma, [12, 28(i)].Of course the local singularity can also be measured at a boundary point x.In this case 3iVis a 3-cell, mdN is a 1-cell spanning it, and the singularity is a type of spanning 1-cell.In this paper we shall consider only imbeddings whose boundary points are all non-singular.Since a singular point must be a vertex in any triangulation of the pair mczM the singular points are always isolated.If m is compact (as it will be from now on) there can therefore be only a finite number of singular points.For the rest of this paper m will be a 2-sphere and M will be the 4-dimensional euclidean space R A \ that is, the 4-sphere punctured at oo.The basic problem 1) This paper follows our announcement [3].We wish to express our thanks to C.H. Giffen for help in the revision.2) These concepts are due to V.