Locking of Dual Closed In n-space a closed m-surface may be locked with a closed n-1 —m-surface, as are the links of a chain. Through neither of them can a sur face one dimension higher than it be laid that does not intersect the other. Examples: A point-couple and a circle enclosing one of its points in a plane. Two linked curves in R,. For the latter, according to Gauss, the double integral 1 (x'—x)(dy de—de dyl 4-Trif +(y'—y)(dz dx'—dx de) -Hie—z)(dxdji'—dy dx') ((x" — Y) -He —zrli .r, y, z being a point of one curve, x'. y, s one of the other, has the value 1. In the case of one curve coiling repeatedly about the other, the order and degree of their interlocking may again be distinguished in accordance with the reflections of the preceding paragraph. Gauss' integral then gives the degree, which may hap pen to be 0 if the order be even. Two cal surfaces transplanted into R. may interlock. Closed m-surfaces, in 2m + 1-space may lock with themselves. They are then said to form knots. The various shapes of knotted curves in common space have been extensively investigated (Listing, Tait, Simony). These researches have been referred to as topology, a word also used synonymously with analysis situs. The simplest knot is the so-called tre foil-knot. It is formed by a curve of the sixth order whose equation in tetrahedral co-ordi nates is given by Brill. It was thought that these knots might be forms of vortex-rings ac counting for the differences of chemical ele ments. There are no knots in 4-space, surface knots first becoming possible in 5-space, and, in their turn, dissolving in 6-space.
Generalized will be called connected with regard to tangent lines, 2-planes, etc., if any two of these can be moved into one another without ceasing to be tangent to the surface. Two spheres in Rs, e.g., are disconnected as to points, connected as to tangents and tangent planes. The op posite is true for the faces of a polyhedron. But we shall assume that the surfaces con sidered possess all these connectivities. With regard to any of them they may be multiply connected. Connectivity as to points (cs) is the special case treated above. It has been re ferred to as cyclosis or periphraxy (Maxwell) in the case of portions of 3-space. The interior of an anchor-ring, e.g., has cs=2.
The c-1 closed curves by means of which we determined the connectivity of a closed sur face in 3-space will lock with other closed curves either in the space enclosed by the sur face or in the exterior. We are thus led to consider the connectivity of the portion (R,— S,) left after subtracting from R, the points of our surface Ss.
Betti's Numbers of a Closed S.— Imagine Sm, if necessary after continuous de formation, placed in m + 1-space. Find the connectivities c 111.-, of the remainder (Rns+s— Sm) with regard to points, lines, 2 planes, . . . m —1-planes. These are Betti's numbers P>, P2 • Pm--1 of the surface (ek.--Pio. This means that in (Rns-H—Sm) there are Pk 1 independent closed k-surfaces with which certain m—k-surfaces within Sm may lock. Obviously, the m— k-surfaces may be deformed out of Sm into the remainder (An+. Sns), while at the same time, and never ceasing to lock with them, the k-surfaces are deformed so as to be on Sm. This shows that Pm--k is at least equal to Pk, and vice versa, so that finally Pm—k = Pk on any closed bilateral m-surf ace without double points. Betti's number P, for a 2-surface is at the same time its connectivity c.
Ruler's Polyhedron The theo rem holds for any division of a spherical sur face into simply connected districts by frontiers bounded by the vertices in which they concur, that v— f +d = 2, v, being the number of vertices, f of frontiers, and d of districts. Such
a map is regular if each district has the same number fd of frontiers, and if an equal number .f, of these concur in each vertex. We have f, 2 and , and f(2fd—faf, + 2.10 = 2f,fd, f dwhere 2 fd—fdfo+2f, must evidently be positive. This gives rise to only five regular maps corre sponding to the regular polyhedral surfaces of the tetrahedron (self-reciprocal), cube and octahedron (reciprocal to each other), dodec ahedron and icosahedron (reciprocal). The regular 4-dimensional polyhedra are found to be six, viz., two self-reciprocal ones bounded by five tetrahedra and 24 octahedra respectively, one bounded by eight cubes reciprocal to one bounded by 16 tetrahedra, and one bounded by 120 dodecahedra reciprocal to one bounded by 600 tetrahedra.
Euler's formula, extended to maps on closed 2-surfaces of connectivity c: v — f + d= 3— c leads to a superior limit for the number of dis tricts that may be adjacent each to each on such a surface. Heffter has investigated under what conditions this limit is actually attained, while H. S. White shows what regular maps (called by him reticulations) are possible for any given c. Generalizing still further, Euler's formula for a map on any m-surface, i.e , a di vision of it into simply connected parts, the dividing m — 1-surfaces again being divided into simply connected partitions, etc., becomes: ns—s 2( — + ( — 1)9(Psys—q 1), ns being the number of q-dimensional parts or partitions on the map. Since for closed m-sur faces we have Pk= Pws—k, if m is even this be comes 2( —1)04=3—Ps-FPI . ; if m is odd, ( (Poincare).
Since no treatise on analysis situs has been published, a few of the main papers on the subject will here be mentioned. W. Dyck (Math. Annalen Vol. 32, p. 457) gives the literature preceding this article (to 1888). The pertinent publications of the savants named above are as follows: Listing, (Der Census raumlicher Complexe> (Gottingen Abhandlungen 1861), and 'Vorstudien zur Topologic) (Gettinger Studien 1847); C. Jor dan,
la deformation des Surfaces) (Liou ville's Journal set-. 2, 11 1866) ; Klein,
Consult also the article on Analysis Situs by M. Dehn in the 'Encycklopedie der Mathematischen Wissenschaf ten,' and R. L. Moore, 'On the Foundations of Plane Analysis Situs.' (Trans. Am. Math. Soc. for 1916). The subject of analysis situs of higher dimensions, especially of 4-space, has been greatly advanced by the following six recent papers by H. Poincari: 'Analysis Situs' (Journal de l'Ecole Polytechni que 1895) ; 'Compliment a l'A. S.' (Proc. London Math. Soc. 1900) ; 'Second compli ment i1 l'A. (Rendiconti del Circolo matematico di Palermo 1899) ; ''Sur certaines surfaces algebriques' (Bull. Soc. Math. de France 1902) ; 'Sur les cycles des surfaces algebriques' (Journal de Math. 1902) ; quieme compliment I l'A. Rendic. Circ. mat. di Palermo 1904).