An Algebraic Theory of Local Knottedness. I

0. Introduction. A. Summary. A significant problem in topology is the classification of wild arcs in S3. The first fundamental paper in this area was written by Fox and Artin in 1948 [13]. There the authors developed rigorous algebraic means for demonstrating the nontameness of arcs. Some time later (1962) Fox and Harrold [14] succeeded in completely classifying a special class of wild arcs, the Wilder arcs. In 1960 Brody [6] developed invariants of infinitely generated modules and used such invariants to distinguish wild knots. Finally, in 1962 ([1], [2]) Alford and Ball constructed a geometric invariant, the penetration index, capable of distinguishing a large class of wild arcs. In this paper we develop algebraic means for distinguishing wild arcs in S3 which go significantly beyond previously known methods. Unlike most of the above authors we consider the problem from a local rather than a global standpoint. The first three sections of this paper are devoted to extending and refining Brody's algebraic techniques [6]. In §IV, a fundamental invariance theorem (IV.B.3) is proven, i.e., (roughly) if p is an isolated singular interior point of an arc k and U is an arbitrary suitably nice neighborhood of p, then the fundamental group U{U—k) of U — k modulo an equivalence relation, called local equivalence, is an invariant of the singularity/?. Thus, the invariants of \~\{U — k) modulo local equivalence, i.e., the module 9JÎ (associated with U{U — k)) modulo an equivalence relation (III.B.2, III.B.7, IV.A.3, IV.C.l), the Jfcth divisor chains of U{U-k) (II.A.4, III.C.3, IV.C.2), the Arth local topologies An{k,p) of U{U — k) (I.D.3, H.A.. 1, H.A.5), etc., also become invariants of the singularity. In §V some applications are considered. First (V.A) the algebraic invariants of Wilder arcs are found to have some pleasant properties (V.A.5). Moreover, the algebraic analogue (V.A.6) to the geometric classification of Wilder arcs [14] is