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Topological persistence for circle valued maps

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Burghelea,  Dan
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Burghelea, D., & Dey, T. K. (2013). Topological persistence for circle valued maps. Discrete & Computational Geometry, 50(1), 69-98. doi:10.1007/s00454-013-9497-x.


Cite as: https://hdl.handle.net/21.11116/0000-0004-663D-6
Abstract
We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle-valued map on an input simplicial complex.