Representation Stability for Configuration Spaces of Graphs
Abstract
We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$ given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an action of the symmetric group $\Sigma_k$ by permuting the $H$summands. We show that the sequence of representations of the symmetric group $H_q(\mathrm{Conf}_n(G_\bullet); \mathbf{Q})$, the homology of the ordered configuration space of these spaces, is representation stable in the sense of Church and Farb. In the case where $G$ and $H$ are trees, we provide a similar result for glueing along arbitrary subtrees instead of the base point. Furthermore, we show that stabilization alway holds for $q = 1$.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.03490
 arXiv:
 arXiv:1701.03490
 Bibcode:
 2017arXiv170103490L
 Keywords:

 Mathematics  Algebraic Topology;
 55R80;
 57M15
 EPrint:
 Major rewrite