2-Complexes with Large 2-Girth
The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in H[subscript 2](X,Z/2) . We consider the maximum possible girth of a complex with n vertices and m 2-faces. If m=n[superscript 2+α] for α<1/2 , then we show that the 2-girth is at most 4n[superscript 2...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2018
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| Online Access: | http://hdl.handle.net/1721.1/117122 https://orcid.org/0000-0002-1302-8657 |
| Summary: | The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in H[subscript 2](X,Z/2) . We consider the maximum possible girth of a complex with n vertices and m 2-faces. If m=n[superscript 2+α] for α<1/2 , then we show that the 2-girth is at most 4n[superscript 2−2α] and we prove the existence of complexes with 2-girth at least c[subscript α,ϵ]n[superscript 2−2α−ϵ]. On the other hand, if α>1/2, the 2-girth is at most Cα . So there is a phase transition as α passes 1 / 2. Our results depend on a new upper bound for the number of combinatorial types of triangulated surfaces with v vertices and f faces. Keywords: Random simplicial complexes, Homology, Counting surfaces |
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