Weighted Fundamental Group
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Weighted Fundamental Group Chengyuan Wu1
· Shiquan Ren2 · Jie Wu3 · Kelin Xia4,5
Received: 29 April 2019 / Revised: 29 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we develop and study the theory of weighted fundamental groups of weighted simplicial complexes. When all weights are 1, the weighted fundamental group reduces to the usual fundamental group as a special case. We also study weighted versions of classical theorems like van Kampen’s theorem. In addition, we also investigate the abelianization, lower central series and applications of weighted fundamental groups. Keywords Algebraic topology · Weighted fundamental group · Applied topology · Weighted simplicial complex · Weighted van Kampen’s theorem Mathematics Subject Classification Primary 55Q05 · 55M99; Secondary 55U10
1 Introduction Weighted structures, such as weighted graphs, are common in mathematics. The addition of weights to a mathematical object often adds new information to the object. Other than weighted graphs, weights have also been considered on hypergraphs [15,18,23] and simplicial complexes [8,22,24,32]. The fundamental group is an important topological invariant. In this paper, our goal is to study the weighted fundamental group of a weighted simplicial complex. Intu-
Communicated by Rosihan M. Ali. Chengyuan Wu, Shiquan Ren, Jie Wu, Kelin Xia—First authors. The project was supported in part by the Singapore Ministry of Education research grant (AcRF Tier 1 WBS No. R-146-000- 222-112). The first author was supported in part by the President’s Graduate Fellowship of National University of Singapore. The second author was supported by the Postdoctoral International Exchange Program of China 2019 project from The Office of China Postdoctoral Council, China Postdoctoral Science Foundation. The third author was supported by Natural Science Foundation of China (NSFC grant no. 11971144) and High-level Scientific Research Foundation of Hebei Province. The fourth author was supported by Nanyang Technological University Startup Grants M4081842, Singapore Ministry of Education Academic Research Fund Tier 1 RG31/18, Tier 2 MOE2018-T2-1-033. Extended author information available on the last page of the article
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itively, the weighted fundamental group should contain information about the weights of the simplicial complex. In addition, the weighted fundamental group should reduce to the usual fundamental group as a special case. Ideally, the weighted fundamental group should also satisfy (weighted versions of) classical theorems like van Kampen’s theorem. We show that our definition of the weighted fundamental group fulfills the above requirements. Our approach is to modify the description of fundamental groups using maximal trees, by introducing weights on edges (1-simplices). To our knowledge, there is no existing literature on weighted fundamental groups of weighted simplicial complexes. In [5], a weighted combinatorial group theory is defined; howe
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