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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spaces of geodesic triangulations of the sphere


Authors: Marwan Awartani and David W. Henderson
Journal: Trans. Amer. Math. Soc. 304 (1987), 721-732
MSC: Primary 57Q15; Secondary 52A55, 57Q37, 58D99
MathSciNet review: 911092
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Abstract: We study questions concerning the homotopy-type of the space $ \operatorname{GT} (K)$ of geodesic triangulations of the standard $ n$-sphere which are (orientation-preserving) isomorphic to $ K$. We find conditions which reduce this question to analogous questions concerning spaces of simplexwise linear embeddings of triangulated $ n$-cells into $ n$-space. These conditions are then applied to the $ 2$-sphere. We show that, for each triangulation $ K$ of the $ 2$-sphere, certain large subspaces of $ \operatorname{GT} (K)$ are deformable (in $ \operatorname{GT} (K)$) into a subsapce homeomorphic to $ \operatorname{SO} (3)$. It is conjectured that (for $ n = 2$) $ \operatorname{GT} (K)$ has the homotopy of $ \operatorname{SO} (3)$. In a later paper the authors hope to use these same conditions to study the homotopy type of spaces of geodesic triangulations of the $ n$-sphere, $ n > 2$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0911092-3
PII: S 0002-9947(1987)0911092-3
Article copyright: © Copyright 1987 American Mathematical Society