Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Additivity of derived limits in the Cohen model

Nathaniel Bannister

Abstract

We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v sák, and Chris Lambie-Hanson which focuses on the system $\mathbf{A}$. In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals and reframe the propagating trivializations results of Bergfalk, Hru\v sák and Lambie-Hanson as a theorem of ZFC. In light of results of the author, Jeffrey Bergfalk, and Justin Moore, the additivity of derived limits also implies additivity results for strong homology.

Additivity of derived limits in the Cohen model

Abstract

We show that, in the model constructed by adding sufficiently many Cohen reals, derived limits are additive on a large class of systems. This generalizes the work of Jeffrey Bergfalk, Michael Hru\v sák, and Chris Lambie-Hanson which focuses on the system . In the process, we isolate a partition principle responsible for the vanishing of derived limits on collections of Cohen reals and reframe the propagating trivializations results of Bergfalk, Hru\v sák and Lambie-Hanson as a theorem of ZFC. In light of results of the author, Jeffrey Bergfalk, and Justin Moore, the additivity of derived limits also implies additivity results for strong homology.
Paper Structure (14 sections, 20 theorems, 62 equations)

This paper contains 14 sections, 20 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notational conventions
  4. Higher Dimensional $\Delta$-systems
  5. $\Omega_{\kappa}$ systems
  6. $\Omega_\kappa$ systems and strong homology
  7. Coherence and Triviality
  8. Injective Systems
  9. Propagating Trivializations for Injective Systems
  10. A Local Partition Result in Cohen Extensions
  11. From Partition to Trivialization
  12. Trivialization on a Large Set
  13. Trivialization Everywhere
  14. Conclusion

Key Result

Corollary 1.1

It is consistent relative to ZFC that strong homology is additive on the class of locally compact separable metric spaces.

Theorems & Definitions (70)

  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 60 more