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Locally classical stable planes

Rainer Löwen

Abstract

We show that a simply connected stable plane with connected lines is isomorphic to an open subplane of a classical projective plane (i.e., a plane over the real or complex numbers, the quaternions or the octonions) if it has that property locally. We also give several examples indicating that our hypotheses cannot be relaxed much further.

Locally classical stable planes

Abstract

We show that a simply connected stable plane with connected lines is isomorphic to an open subplane of a classical projective plane (i.e., a plane over the real or complex numbers, the quaternions or the octonions) if it has that property locally. We also give several examples indicating that our hypotheses cannot be relaxed much further.
Paper Structure (6 sections, 8 theorems)

This paper contains 6 sections, 8 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homomorphisms and local homomorphisms
  4. Isotopies of arcs
  5. Main results
  6. Examples

Key Result

LEMMA 3.1

Every homomorphism of stable planes is injective.

Theorems & Definitions (16)

  • LEMMA 3.1
  • DEFINITION 3.2
  • LEMMA 3.3
  • LEMMA 3.4
  • LEMMA 3.5
  • THEOREM 3.6
  • THEOREM 4.1
  • THEOREM 5.1
  • COROLLARY 5.2
  • Example 6.1
  • ...and 6 more