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:
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arXiv e-prints
- Pub Date:
- January 2017
- DOI:
- 10.48550/arXiv.1701.03490
- arXiv:
- arXiv:1701.03490
- Bibcode:
- 2017arXiv170103490L
- Keywords:
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- Mathematics - Algebraic Topology;
- 55R80;
- 57M15
- E-Print:
- Major rewrite