Euclidean Domains With Uniformly Abelian Local Fundamental Groups
O. G. Harrold · Transactions of the American Mathematical Society · 1949
If C and C1 are homeomorphic closed subsets of the Euclidean three space, R3, a necessary condition that their complementary sets be homeomorphic is that the fundamental groups of their residual sets be isomorphic.When these groups are not isomorphic the situation may be described roughly by saying that C and C1 are imbedded differently in R3 either in the large, or in the small, or both.Examples of the first situation arise if C and C1 are polygonal simple closed curves, one knotted, the other unknotted.Examples of the second variety arise if C and C1 are taken to be an ordinary linear interval and an arc of Antoine [2](x) respectively.In this note we are interested in establishing conditions on the complement of certain classes of homeomorphic closed sets such that the corresponding fundamental groups will vanish.A general problem, which is solved below only in special cases indeed, is: If C is an absolute retract in the «-sphere, Sn, under what conditions does Ti(S*-C) vanish?In case Cis a topological î'-cell, i = l, 2, ■ • -, », it is sufficient that Sn -C have uniformly abelian local fundamental groups (Theorem IV).The author wishes to express his appreciation to S. Eilenberg, whose helpful criticisms have made him practically a collaborator.2. Let X be a topological space, A a subset of X.We shall consider the 1-dimensional homology groups Hx(A) and Hx(X) based on singular homologies with integer coefficients.The identity map i:A-+X induces a homomorphism i*:Hx(A)->(HxX).The image of this homomorphism will be denoted by Hx(A, X) and is the homology group obtained by considering 1-cycles in A and bounding in X (not to be confused with the relative homology group Hx(X, A)).Assume that both X and A are arcwise connected and let pÇîA be the base point used to define the fundamental groups 7Ti(.4) and Tx(X).Again the identity map i'.A-=>X induces a homomorphism if.wx(A)-+ttx(X).The image of this homomorphism will be denoted by 7ti(.4,X).Up to an isomorphism, this group is independent of the choice of pÇ.A. Clearly Hx(A, X) and 7ri(^4, X) are subgroups of Hx(X) and irx(X), respectively.If BCA, then Hx(B, X) C Hx(A, X) and vt(B, X) C Vt{A, X).