On the second homology of planar graph braid groups
Byung Hee An, Ben Knudsen
TL;DR
This paper investigates the second homology of configuration spaces of planar graphs, demonstrating it is generated by three atomic graphs under specific operations, with a counterexample for non-planar graphs.
Contribution
It identifies the generators of the second homology for planar graph configuration spaces and highlights the distinction with non-planar graphs.
Findings
Second homology of planar graph configuration spaces is generated by three atomic graphs.
Embedding, disjoint union, and edge stabilization are key operations.
A non-planar graph example shows the result does not extend universally.
Abstract
We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Algebraic structures and combinatorial models