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Discrete & Computational Geometry

, Volume 62, Issue 4, pp 912–944 | Cite as

Configuration Spaces of Graphs with Certain Permitted Collisions

  • Eric RamosEmail author
Article
  • 29 Downloads

Abstract

If G is a graph with vertex set V, let \({{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)\) be the space of n-tuples of points on G, which are only allowed to overlap on elements of V. We think of \({{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)\) as a configuration space of points on G, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of \({{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)\) in the case where G is a tree. Next, we use techniques of asymptotic algebra to prove statements about \({{\mathrm{Conf}}}_n^{{{\mathrm{sink}}}}(G,V)\), for general graphs G, whenever n is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of Ramos (arXiv:1606.02673, 2016).

Keywords

FI-modules Representation stability Configuration spaces of graphs 

Mathematics Subject Classification

05C10 16G20 18A25 

Notes

Acknowledgements

The author would like to send thanks to Jordan Ellenberg, John Wiltshire-Gordon, Jennifer Wilson and Graham White for various useful conversations during the writing and conception of this work. The author would also like to send very special thanks to Steven Sam for his support during the initial stages of this work. The author would like to send thanks to the two anonymous referees, whose various suggestions greatly improved the overall quality of this work. Finally, the author would like to acknowledge the generous support of the NSF via the grant DMS-1502553.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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