Free σ-products and fundamental groups of subspaces of the plane

Let H be the so-called Hawaiian earring, i.e., H = {(x, y): (x - 1/n)(2) + y(2) = 1/n(2), 1 less than or equal to n < omega} and o = (0, 0). We prove: (1) If Y is a subspace of a line in the Euclidean plane R-2 and X its complement R-2 \ Y With x is an element of X, then the fundamental group pi(1)(X, x) is isomorphic to a subgroup of pi(1)(H-1 o). (2) Let Y be a subspace of a line in the Euclidean plane R-2. Then, pi(1)(R-2 \ Y, x) for x is an element of R-2 \ Y is isomorphic to pi(1)(H, o), if and only if there exists infinitely many connected components of Y which converge to a point outside of Y. (3) Every homomorphism from pi(1)(H, o) to itself is conjugate to a homomorphism induced from a continuous map. (C) 1998 Elsevier Science B.V.