Volume 21, issue 7 (2021)

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Deletion and contraction in configuration spaces of graphs

Sanjana Agarwal, Maya Banks, Nir Gadish and Dane Miyata

Algebraic & Geometric Topology 21 (2021) 3663–3674

Our aim is to provide space level maps between configuration spaces of graphs that are predicted by algebraic manipulations of cellular chains. More explicitly, we consider edge contraction and half-edge deletion, and identify the homotopy cofibers in terms of configuration spaces of simpler graphs. The construction’s main benefit lies in making the operations functorial — in particular, graph minors give rise to compatible maps at the level of fundamental groups as well as generalized (co)homology theories.

As applications we provide a long exact sequence for half-edge deletion in any generalized cohomology theory, compatible with cohomology operations such as the Steenrod and Adams operations, allowing for inductive calculations in this general context. We also show that the generalized homology of unordered configuration spaces is finitely generated as a representation of the opposite graph-minor category.

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configuration spaces, graphs, graph minor, edge contraction, edge deletion, homology, cohomology, homotopy cofiber, mapping cone, braid groups
Mathematical Subject Classification
Primary: 55R80
Secondary: 05C10, 20F36
Received: 20 July 2020
Revised: 1 November 2020
Accepted: 11 December 2020
Published: 28 December 2021
Sanjana Agarwal
Department of Mathematics
Indiana University
Bloomington, IN
United States
Maya Banks
Department of Mathematics
University of Wisconsin
Madison, WI
United States
Nir Gadish
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Dane Miyata
Department of Mathematics
University of Oregon
Eugene, OR
United States