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PreprintCC BY 4.0arXiv

Eilenberg-MacLane spaces in A-homotopy theory

  • Georg Wille
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Abstract

In this paper we study a discrete homotopy theory for undirected graphs. More precisely, we consider A-homotopy groups, first defined in the work of Barcelo, Kramer, Laubenbacher and Weaver. For every degree n and every group G (Abelian if n≥ 2), we construct a graph whose n-th A-homotopy group is isomorphic to G and whose other A-homotopy groups all vanish. Using these, we show that for every infinite sequence of groups G1,G2,…, with Gi Abelian for i≥ 2, there is a graph X such that the i-th A-homotopy group of X is isomorphic to Gi for all i. Along the way, we identify relative A-homotopy groups with relative classical homotopy groups, and prove a discrete version of the Blakers-Massey theorem. We then construct partial Postnikov systems and (full) Whitehead towers.

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Cite

BibTeXRIS
BibTeX
@misc{wille2026eilenbergmaclane,
  title = {{Eilenberg-MacLane spaces in A-homotopy theory}},
  author = {Georg Wille},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03497},
  eprint = {2610.03497},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03497},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Eilenberg-MacLane spaces in A-homotopy theory
AU  - Georg Wille
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03497
UR  - https://arxiv.org/abs/2610.03497
AB  - In this paper we study a discrete homotopy theory for undirected graphs. More precisely, we consider A-homotopy groups, first defined in the work of Barcelo, Kramer, Laubenbacher and Weaver. For every degree n and every group G (Abelian if n≥ 2), we construct a graph whose n-th A-homotopy group is isomorphic to G and whose other A-homotopy groups all vanish. Using these, we show that for every infinite sequence of groups G_1,G_2,…, with G_i Abelian for i≥ 2, there is a graph X such that the i-th A-homotopy group of X is isomorphic to G_i for all i. Along the way, we identify relative A-homotopy groups with relative classical homotopy groups, and prove a discrete version of the Blakers-Massey theorem. We then construct partial Postnikov systems and (full) Whitehead towers.
ER  -