A locally simply connected space and fundamental groups of one point unions of cones
A locally simply connected space and fundamental groups of one point unions of cones
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局部单连通空间和圆锥单点并的基本群
DOI：
10.1090/s0002-9939-1992-1132409-0
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
K. Eda
中科院分区：
文献类型：
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作者：
K. Eda
设Cx是空间X上的锥体，设X在x处第一可数，则下列条件等价：(1)X在x处局部单连通；(2)IR1((X，x)V(X，x)，x)自然同构于自由积ri(X，x)*r1(X，x)；(3)ri((cx，x)V(cx，x)，x)是平凡的。存在一个单连通的局部单连通的Tychonoff空间X和xeX，使得(X，x)V(X，x)不是单连通的。Griffiths[2]证明了定理1(H.B.Griffiths)。设空间X在x E X处局部单连通，且在x处第一可数，则对任意y E Y空间Y，一点并(X，x)V(Y，y)的基本群7ri((X，x)v(Y，y)，x)自然同构于自由积7r1(X，x)*7r，(Y，y).此外，在同一篇文章中，他还证明了局部单连通性在这个定理中是本质的。另一方面，本文的作者[1]证明了定理中的第一可数性也是必要的。在？1中，我们将证明定理2。设空间X，Y分别在x，y处是第一可数的。(2)7rI((X，x)V(Y，y)，x)自然同构于自由积rl(X，X)*7fl(Y，Y)；(3)7rI((cx，x)V(Cy，y)，x)是平凡的，其中Cx，Cy分别是X，Y上的锥体。推论3.设X在x处第一可数，则下列条件等价：(1)X在x处局部单连通；(2)7R1((X，x)V(X，x)，x)自然同构于自由积ILR(X，x)*7RL(X，x)；(3)ri((Cx，x)V(Cx，x)，x)是平凡的。编辑于1990年2月27日收到，并以修订后的形式于1990年11月26日收到。1991年数学科目分类。主55Q20、55Q52。
Let CX be the cone over a space X. Let a space X be first countable at x, then the following are equivalent: (1) X is locally simply connected at x; (2) ir1 ((X, x) V (X, x), x) is naturally isomorphic to the free product r I(X, x) * r1(X, x); (3) r I((CX, x) V (CX, x), x) is trivial. There exists a simply connected, locally simply connected Tychonoff space X with x e X, such that (X, x) V (X, x) is not simply connected. Griffiths [2] proved Theorem 1 (H. B. Griffiths). Let a space X be locally simply connected at x E X and also first countable at x. Then, for an arbitrary space Y with y E Y, the fundamental group 7r I((X, x) v (Y, y), x) of the one point union (X, x) V (Y, y) is naturally isomorphic to the free product 7r1 (X, x) * 7r,(Y, y). In addition, in the same paper he proved that local simple connectivity is essential in this theorem. On the other hand the author [1] of the present paper has shown that the first countability in the theorem is also essential. In ? 1 we shall prove Theorem 2. Let spaces X, Y be first countable at x, y, respectively. Then the following are equivalent: (1) At least one of X and Y is locally simply connected at x or y, respectively; (2) 7rI((X, x) V (Y, y), x) is naturally isomorphic to the free product rl (X, X) * 7fl(Y, Y) ; (3) 7rI ((CX, x) V (CY, y), x) is trivial, where CX, CY are the cones over X, Y, respectively. Corollary 3. Let a space X be first countable at x. Then the following are equivalent: (1) X is locally simply connected at x; (2) 7r1 ((X, x) V (X, x), x) is naturally isomorphic to the free product ilr(X, x) * 7rl(X, x); (3) r I((CX, x) V (CX, x), x) is trivial. Received by the editors February 27, 1990 and, in revised form, November 26, 1990. 1991 Mathematics Subject Classification. Primary 55Q20, 55Q52.