A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

2010/06/18 by Mao, Linfan
#51H20 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.1006.4071

Abstract

As we known, the \it Seifert-Van Kampen theorem handles fundamental groups of those topological spaces X=U∪ V for open subsets U, V⊂ X such that U∩ V is arcwise connected. In this paper, this theorem is generalized to such a case of maybe not arcwise-connected, i.e., there are C₁, C₂,..., C_m arcwise-connected components in U∩ V for an integer m≥ 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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