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PreprintarXiv licencearXiv

Brown's Asymptotic Limit Functor and Proper Homology

  • Sumanta Das
  • Rekha Santhanam
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Abstract

In 1974, E. M. Brown introduced the ℘-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that ℘ has major advantages over the classical limits varprojlim and varprojlim1: it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown--Grossman proper fundamental group of any open contractible $3$-manifold other than ℝ3 is uncountable and perfect.

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Cite

BibTeXRIS
BibTeX
@misc{das2026browns,
  title = {{Brown's Asymptotic Limit Functor and Proper Homology}},
  author = {Sumanta Das and Rekha Santhanam},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14063},
  eprint = {2609.14063},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14063},
}
RIS
TY  - UNPB
TI  - Brown's Asymptotic Limit Functor and Proper Homology
AU  - Sumanta Das
AU  - Rekha Santhanam
PY  - 2026
DA  - 2026/09/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.14063
UR  - https://arxiv.org/abs/2609.14063
AB  - In 1974, E. M. Brown introduced the ℘-functor to study the proper homotopy groups of an end, suggesting a parallel proper homology theory that has since remained undeveloped. In this paper, we construct this missing proper homology and relate it to proper homotopy by developing a proper Hurewicz theorem. Bypassing abstract pro-categorical machinery, we show that ℘ has major advantages over the classical limits varprojlim and varprojlim^1: it is exact, detects pro-triviality, satisfies a cardinality dichotomy, and connects to these classical limits through a 4-term exact sequence. As an application, we prove that the Brown--Grossman proper fundamental group of any open contractible $3$-manifold other than ℝ^3 is uncountable and perfect.
ER  -