ANALYSIS SITUS. Let a geometrical figure — say a closed surface in common space —be subjected to any change of form (bend ing, stretching, etc.) that does not involve any or An extensible rubber model will suggest the possibilities of such de formation. Whatever properties of our figure are unalterable by this process from the subject matter of analysis situs, which may therefore be defined as the theory of invariants of the group (or groups, see this term) of continuous de formations. Its scope, however, is not con fined to common space, but embraces, in gen eral, m-dimensional figures in n-dimensional space (more briefly: Rn for n-space, also m surfaces for m-dimensional surfaces, etc.).
The effect of tearing a surface or making an incision on it along a line is to double the lat ter. As the incision proceeds it substitutes for each point P of the line two points, Pi, Pr, henceforth not to be considered as consecutive, and whose successions separately constitute the left and right edges of the incision. Joining is the opposite process, each point of the junc ture consisting of twin points merged into one. Corresponding definitions apply to incision and juncture along surfaces of two or more dimen sions, or when the elements considered are straights, planes, etc., instead of points.
It will here be noticed that figures which are not continuously deformable into one an other, or equivalent, in n-space, may become so by virtue of the additional freedom of defor ation that n 1-space affords. The figure of two concentric circles in a plane is not equiva lent to two circles excluding one another, but becomes so in 3-space. Hence a distinction arises between absolute analysis situs, which places its figures in space of any suitable num ber of dimensions, and analysis situs in a given space or surface within which all deformation must take place.
C. Jordan has shown that in the case of 2 dimensional surfaces the following four inva riants form a complete system. This means that any two surfaces agreeing in these data are equivalent: (1) the number of detached por tions of which they consist, and, with regard to each of these: (2) the number of curves bounding it; (3) its connectivity; (4) its later ality (unilateral or bilateral type). Evidently the first and second of these could be changed by incision or juncture only.
Connectivity.— A surface is connected if it permits of continuous passage on it between any two of its points. The standard of connec tivity is the area of a plane triangle, circle or equivalent figure, which is called simply-con nected or elementary. On it any two curves C., C. (not intersecting themselves or each other) between two points, A, B, are equiva lent, and taken together they form a closed curve which divides the plane into two separate portions. This latter property received analyti cal demonstration from Jordan (hence "Jordan curves))) and has lately been based on the theory of assemblages by Veblen. Using Poincare's notation we write ==*- C2 or Ci — Cs 0, where the negative sign means that the curve is to be taken in the opposite direction (from B to A), and equivalence to zero means un limited contractibility. A spherical or ellipsoidal surface is also simply connected, with this dif ference, that closed curves, if one obstacle (a small circle or "puncture' of the surface) be placed in the way of their contraction, may still be reduced to zero by deformation in the oppo site direction. Consider, for instance, the in tersection of such a surface with a movable plane as the latter moves parallel to itself in either of two directions.
Extending our definition of equivalence to zero, to sums of curves on any surface, it be comes necessary to stipulate (1) that the order of terms of a sum must he preserved (non commutative addition) and (2) that any por tions of curves, if deformed so as to coincide and form negatives of one another, shall cancel. Thus on the surface of the double ring (Fig. 1) we have Ci Ci= Ca or C, A- C.— Cs E0. Curves form an independent set on a surface if none and no sums of them are equivalent to zero. Curves containing portions equivalent inter se (as when coiling several times about a cylinder) shall here be excluded. Multiply-con niTted surfaces are then said to have connec tivity c if they permit of c independent paths between any two points A, B. The connectivity of a closed surface, i.e., one without boundary and yet having all its points at finite distances, is not changed by puncturing it. For instance, the intersection of the double ring of Fig. 1 with a 5lane remains equivalent with itself (and to zero) no matter how the plane moves.