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Configuration Spaces of Graphs with Certain Permitted Collisions

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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).

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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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  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Eric Ramos

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  1. Eric Ramos
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Correspondence to Eric Ramos.

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Editor in Charge: Kenneth Clarkson

The author was supported by NSF grant DMS-1502553.

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Ramos, E. Configuration Spaces of Graphs with Certain Permitted Collisions. Discrete Comput Geom 62, 912–944 (2019). https://doi.org/10.1007/s00454-018-0045-6

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