Alphabeta MathSmall-Cancellation Disc Diagrams and the Torsion Toolkit
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1 · Prerequisites
- Binary Operations, Monoids, Groups and Subgroups
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Finite Counting, Factorials and Binomial Coefficients
- Free Groups and Presentations
- Normal Subgroups and Quotient Groups
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
2 · Logical flowchart
3 · Items
Definition
Sc toolkit symmetrised relators and pieces
Definition
Sc toolkit labelled planar disc diagram
Theorem
Sc toolkit van kampen existence
Lemma
Sc toolkit minimal diagrams and cut vertex reduction
Definition
Arc reduction and combinatorial curvature of a disc diagram
Lemma
Internal arcs of a reduced small cancellation diagram are pieces
Lemma
C prime one sixth interior faces have at least seven arcs
Lemma
Euler curvature identity for an arc reduced disc diagram
Lemma
Boundary spur or at most three shell from curvature
Theorem
Greendlinger shell existence from the curvature count
Definition
Minimal cyclic power diagram and relator root
Lemma
Sc toolkit commuting positive words have a common root
Lemma
Sc toolkit periodic relator overlap is a piece
Definition
Cyclically Dehn-reduced words
Lemma
Periodic words: a relator root or Dehn-reduced powers
Lemma
A shortest finite-order representative shares a word root with a relator
Theorem
C prime one sixth torsion elements come from relator roots
Sources
- Touikan §3.5 Definition 3.5.2 (comparison of conventions)
- Lipschutz (1964), §2, printed pp.37–38: cancellation condition with the identical-word exception
- Touikan §3.4 opening definitions
- Touikan §3.1, Theorem 3.1.7 and the balloon-diagram construction
- Touikan §3.5 Figure 3.5.1 and nonsingular restriction before Definition 3.5.3
- Touikan §3.4.2 curvature formulas; §3.5.1 arc reduction
- Touikan §3.5 Figure 3.5.1 and following paragraph
- Touikan §3.5.1 paragraph preceding Definition 3.5.3
- Touikan Lemmas 3.4.2–3.4.3, Corollary 3.4.5 and Theorem 3.4.6
- Touikan Proposition 3.5.5, complete proof
- Touikan Corollary 3.5.8, with singular-diagram obligation carried from the preceding lemma
- Lipschutz (1964), §2 and §6; minimum-length choices expanded locally
- Lipschutz (1964), §6 opening periodic-word argument; commuting-word input expanded locally
- Lipschutz (1964), §6 opening remark, printed p.41
- Lipschutz (1964), §2, printed pp.37–38
- Lipschutz (1964), Lemma 4 and complete §6 proof, printed pp.39,41–42; common-root exceptions retained locally
- Lipschutz (1964), §6, printed pp.41–42; local torsion deduction retaining the common-root alternative
- Lipschutz (1964), §6 periodic-word proof, with the local torsion deduction