In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closedhyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.
With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a (finite volume) quotient space of the (non-finite volume) order-5 dodecahedral honeycomb, a regulartessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
Regina – Support Data: Weber-Seifert dodecahedral space
The Weber–Seifert dodecahedral space: Answering a computational challenge
January 26, 2023
seifert, weber, space, mathematics, introduced, herbert, seifert, constantin, weber, closed, hyperbolic, manifold, also, known, seifert, weber, dodecahedral, space, hyperbolic, dodecahedral, space, first, discovered, examples, closed, hyperbolic, manifolds, co. In mathematics Seifert Weber space introduced by Herbert Seifert and Constantin Weber is a closed hyperbolic 3 manifold It is also known as Seifert Weber dodecahedral space and hyperbolic dodecahedral space It is one of the first discovered examples of closed hyperbolic 3 manifolds It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3 manifold There are three ways to do this gluing consistently Opposite faces are misaligned by 1 10 of a turn so to match them they must be rotated by 1 10 3 10 or 5 10 turn a rotation of 3 10 gives the Seifert Weber space Rotation of 1 10 gives the Poincare homology sphere and rotation by 5 10 gives 3 dimensional real projective space With the 3 10 turn gluing pattern the edges of the original dodecahedron are glued to each other in groups of five Thus in the Seifert Weber space each edge is surrounded by five pentagonal faces and the dihedral angle between these pentagons is 72 This does not match the 117 dihedral angle of a regular dodecahedron in Euclidean space but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60 and 117 and the hyperbolic dodecahedron with dihedral angle 72 may be used to give the Seifert Weber space a geometric structure as a hyperbolic manifold It is a finite volume quotient space of the non finite volume order 5 dodecahedral honeycomb a regular tessellation of hyperbolic 3 space by dodecahedra with this dihedral angle The Seifert Weber space is a rational homology sphere and its first homology group is isomorphic to Z 5 3 displaystyle mathbb Z 5 3 William Thurston conjectured that the Seifert Weber space is not a Haken manifold that is it does not contain any incompressible surfaces Burton Rubinstein amp Tillmann 2012 proved the conjecture with the aid of their computer software Regina References EditBarbieri Elena Cavicchioli Alberto Spaggiari Fulvia 2009 Some series of honey comb spaces The Rocky Mountain Journal of Mathematics 39 2 381 398 Weber Constantin Seifert Herbert 1933 Die beiden Dodekaederraume Mathematische Zeitschrift 37 1 237 253 doi 10 1007 BF01474572 MR 1545392 Thurston William 1997 Levy Silvio ed Three dimensional geometry and topology Vol 1 Princeton Mathematical Series vol 35 Princeton NJ Princeton University Press ISBN 0 691 08304 5 Burton Benjamin A Rubinstein J Hyam Tillmann Stephan 2012 The Weber Seifert dodecahedral space is non Haken Transactions of the American Mathematical Society 364 911 932 arXiv 0909 4625 doi 10 1090 S0002 9947 2011 05419 X Weeks Jeffrey The shape of space 2nd ed Marcel Dekker pp 219 ISBN 978 0824707095 External links EditRegina Support Data Weber Seifert dodecahedral space The Weber Seifert dodecahedral space Answering a computational challenge Retrieved from https en wikipedia org w index php title Seifert Weber space amp oldid 1095484933, wikipedia, wiki, book, books, library,
