A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

Details A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

2010-06-18

arxiv.org/abs/1006.4071v1

Mathematics

General Mathematics

16 pages with 3 figures

Scientific paper

As we known, the {\it Seifert-Van Kampen theorem} handles fundamental groups of those topological spaces $X=U\cup V$ for open subsets $U, V\subset X$ such that $U\cap V$ is arcwise connected. In this paper, this theorem is generalized to such a case of maybe not arcwise-connected, i.e., there are $C_1$, $C_2$,$..., C_m$ arcwise-connected components in $U\cap V$ for an integer $m\geq 1$, which enables one to find fundamental groups of combinatorial spaces by that of spaces with theirs underlying topological graphs, particularly, that of compact manifolds by their underlying graphs of charts.

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