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First countability and local simple connectivity of one point unions. (English) Zbl 0697.55016

H. B. Griffiths [Q. J. Math., Oxf., II. Ser. 5, 175-190 (1954; Zbl 0056.163)] proved that if X is path-connected, locally simply connected at \(x\in X\) and also first countable at x, then for an arbitrary path- connected space Y with \(y\in Y\), the fundamental group of the one point union (X,x)\(\vee (Y,y)\) is naturally isomorphic to the free product \(\pi\) (X)*\(\pi\) (Y). The author shows that the assumption of first countability is essential by exhibiting a path-connected Tychonoff space X with \(x\in X\) such that X is locally simply connected at x, \(\pi\) (X) is trivial but \(\pi\) ((X,x)\(\vee (X,x))\) is not trivial.
Reviewer: S.C.Althoen

MSC:

55Q20 Homotopy groups of wedges, joins, and simple spaces
55Q05 Homotopy groups, general; sets of homotopy classes
57M05 Fundamental group, presentations, free differential calculus

Citations:

Zbl 0056.163
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References:

[1] H. B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math., Oxford Ser. (2) 5 (1954), 175 – 190. · Zbl 0056.16301 · doi:10.1093/qmath/5.1.175
[2] Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. · Zbl 0088.38803
[3] John W. Morgan and Ian Morrison, A van Kampen theorem for weak joins, Proc. London Math. Soc. (3) 53 (1986), no. 3, 562 – 576. · Zbl 0609.57002 · doi:10.1112/plms/s3-53.3.562
[4] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. · Zbl 0145.43303
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