The homology of configuration spaces of trees with loops

@article{Chettih2016TheHO,
  title={The homology of configuration spaces of trees with loops},
  author={Safia Chettih and Daniel Lutgehetmann},
  journal={Algebraic \& Geometric Topology},
  year={2016},
  volume={18},
  pages={2443-2469}
}
We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the $E^1$-page and differentials of Mayer… 

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References

SHOWING 1-10 OF 12 REFERENCES

Homology groups for particles on one-connected graphs

We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for

Topology of configuration space of two particles on a graph, II

This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the

Topology of Configuration Space of Two Particles on a Graph, I

In this paper we study the homology and cohomology of confguration spaces of two distinct particles on a graph. Our main tool is intersection theory for cycles in graphs. We obtain an explicit

Estimates for homological dimension of configuration spaces of graphs

We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is

Metric Spaces of Non-Positive Curvature

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by

Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups

We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding

Finding Topology in a Factory: Configuration Spaces

A class of topological spaces related to motion-planning on graphs that arise naturally in this very context, arising simultaneously in two seemingly disparate fields are described.