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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A locally simply connected space and fundamental groups of one point unions of cones


Author: Katsuya Eda
Journal: Proc. Amer. Math. Soc. 116 (1992), 239-249
MSC: Primary 55Q20; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9939-1992-1132409-0
MathSciNet review: 1132409
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Abstract: 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) ${\pi _1}\left ( {\left ( {X,x} \right ) \vee \left ( {X,x} \right ),x} \right )$ is naturally isomorphic to the free product ${\pi _1}\left ( {X,x} \right ) * {\pi _1}\left ( {X,x} \right )$; (3) ${\pi _1}\left ( {\left ( {CX,x} \right ) \vee \left ( {CX,x} \right ),x} \right )$ is trivial. There exists a simply connected, locally simply connected Tychonoff space $X$ with $x \in X$, such that $\left ( {X,x} \right ) \vee \left ( {X,x} \right )$ is not simply connected.


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Keywords: Fundamental group, locally simple, first countable, one point union, cone
Article copyright: © Copyright 1992 American Mathematical Society