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A torus bundle, in the sub-field of geometric topology in mathematics, is a kind of surface bundle over the circle, which in turn is a class of three-manifolds.
which fixes every point of the torus) then the resulting torusbundle M ( f ) {\displaystyle M(f)} is the three-torus: the Cartesian product of three...
twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the...
2-torusbundles for trace −2 automorphisms of the 2-torus. For b=−2 this is an oriented Euclidean 2-torusbundle over the circle (the surface bundle associated...
(trivial) bundle is the 2-torus, S 1 × S 1 {\displaystyle S^{1}\times S^{1}} . A covering space is a fiber bundle such that the bundle projection is a local...
nilmanifold, which is the total space of a principal torusbundle over a principal torusbundle over a torus. Hermann Karcher. Report on M. Gromov's almost...
infinity"). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product...
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian...
A simple special case of this construction (considered in Henri Poincaré's foundational paper) is that of a torusbundle. Virtually fibered conjecture...
mapping class group of the two-torus that only lens spaces have splittings of genus one. Three-torus Recall that the three-torus T 3 {\displaystyle T^{3}}...
3t\end{aligned}}} The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus ( r − 2 ) 2 + z 2 = 1...
William Floyd and Allen Hatcher, Incompressible surfaces in punctured-torusbundles, Topology and its Applications 13 (1982), no. 3, 263–282. Allen Hatcher...
tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension n {\displaystyle n} is also parallelizable, as can be seen...
surface. When the base space is a circle the total space is three-dimensional and is often called a surface bundle over the circle. Mapping torus v t e...
(D^{n+1})\simeq \operatorname {BTop} (S^{n}).} An example of a sphere bundle is the torus, which is orientable and has S 1 {\displaystyle S^{1}} fibers over...
(especially Picard varieties and Albanese varieties). A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex...
spaces and bundles over projective space. The original motivation to study toric varieties was to study torus embeddings. Given the algebraic torus T, the...
represents that line bundle over the torus. If one takes the skyscraper sheaf supported on that point in the dual torus, then we see torus fibres of the SYZ...
image of the other, yield a fundamental region of the torus. The universal cover of both the torus and the Klein bottle is the plane R2. The fundamental...
projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally...
In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f...
embeddings, f and g, of the torus into R3. X is the torus, Y is R3, f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut...
that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an...
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