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A decomposition of $E^3$ into straight arcs and singletons

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Armentrout S.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction............................................................................................................................................. 5 2. Notation and. terminology................................................................................................................... 6 3. Description of G.................................................................................................................................... 6 4. Preliminaries to the main result........................................................................................................ 13 5. The main result..................................................................................................................................... 15 6. Lemmas on Property X......................................................................................................................... 16 7. Construction of certain open sets...................................................................................................... 20 8. Constructing homotopy centerlines.................................................................................................. 24 9. Patterns on the sides of Δ.................................................................................................................. 28 10. The AB-condition................................................................................................................................. 30 11. Types of simple closed curves......................................................................................................... 32 12. Similar patterns................................................................................................................................... 36 13. Property II.............................................................................................................................................. 40 14. Construction of homotopy-$h[Γ_α]'s$............................................................................................. 42 15. Proof of Lemma 3................................................................................................................................ 45 References.................................................................................................................................................. 46

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1 Armentrout S.

1 1970

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A decomposition of $E^3$ into straight arcs and singletons