A Generalization of Seifert-Van Kampen Theorem for Fundamental Groups

zbMATH Open1252.57003arXiv1006.4071MaRDI QIDQ4648366

Publication date: 9 November 2012

Abstract: As we known, the {it Seifert-Van Kampen theorem} handles fundamental groups of those topological spaces

X = U c u p V

for open subsets

U, V s u b s e t X

such that

U c a p 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 c a p V

for an integer

m g e q 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.

Full work available at URL: https://arxiv.org/abs/1006.4071

zbMATH Keywords

fundamental group combinatorial manifold Seifert-Van Kampen theorem topological space topological graph

Mathematics Subject Classification ID

Fundamental group, presentations, free differential calculus (57M05) Homotopy groups (55Q99)