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SPACES OF SPECIAL QUADRILATERALS

  • AHTZIRI GONZÁLEZ (a1) and JORGE L. LÓPEZ-LÓPEZ (a2)
Abstract

We describe the parameter spaces of some families of quadrilaterals, such as parallelograms, rectangles, rhombuses, cyclic quadrilaterals and trapezoids. For this purpose, we prove that the closed $n$ -disc $\mathbb{D}^{n}$ is the unique topological $n$ -manifold (with boundary) whose boundary and interior are homeomorphic to $\mathbb{S}^{n-1}$ and $\mathbb{R}^{n}$ , respectively. Roughly speaking, our main result states that the natural compactifications of the parameter spaces of cyclic quadrilaterals and of trapezoids, modulo similarity, are both homeomorphic to $\mathbb{D}^{3}$ .

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The first author is partially supported by CONACYT grant FORDECYT 265667; the second author is partially supported by funding from CIC-UMSNH.

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[1] Brown, M., ‘A proof of the generalized Schöenflies theorem’, Bull. Amer. Math. Soc. 66(2) (1960), 7476.
[2] Brown, M., ‘Locally flat imbeddings of topological manifolds’, Ann. of Math. (2) 75(2) (1962), 331341.
[3] González, A. and López-López, J. L., ‘Compactness of spaces of convex and simple quadrilaterals’, Bull. Aust. Math. Soc. 94(3) (2016), 507521.
[4] Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, 2002).
[5] Munkres, J. R., Topology (Prentice Hall, Upper Saddle River, NJ, 2000).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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