which fixes every point of the torus) then the resulting torus bundle M ( f ) {\displaystyle M(f)} is the three-torus: the Cartesian product of three...

is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis...

2-torus bundles for trace −2 automorphisms of the 2-torus. For b=−2 this is an oriented Euclidean 2-torus bundle over the circle (the surface bundle associated...

nilmanifold, which is the total space of a principal torus bundle over a principal torus bundle over a torus. Hermann Karcher. Report on M. Gromov's almost...

A simple special case of this construction (considered in Henri Poincaré's foundational paper) is that of a torus bundle. Virtually fibered conjecture...

torus. It has been shown that every principal torus bundle over a torus is of this form, see. More generally, a compact nilmanifold is a torus bundle...

(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...

I-bundles Knot and link complements Lens space Seifert fiber spaces, Circle bundles Spherical 3-manifold Surface bundles over the circle Torus bundle A...

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...

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...

pseudo-Anosov. The case where S is a torus (i.e., a surface whose genus is one) is handled separately (see torus bundle) and was known before Thurston's work...

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}}...

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...

Examples are the 3-torus, and more generally the mapping torus of a finite-order automorphism of the 2-torus; see torus bundle. There are exactly 10...

decomposition Branched surface Lamination Examples 3-sphere Torus bundles Surface bundles over the circle Graph manifolds Knot complements Whitehead manifold...

Allen Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. In a 1980 paper Floyd introduced a way to compactify...

In mathematics, specifically in topology, the mapping torus of a homeomorphism f of some topological space X to itself is a particular geometric construction...

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...

maximal torus H. It extends to a Borel subgroup λ:B→C, giving a one dimensional representation Wλ of B. Then GxWλ is a trivial vector bundle over G on...

real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus. Explicitly, a line bundle on T =...

provided a proof of the majority of the conjecture for nonsingular torus bundles over affine manifolds using ideas from the SYZ conjecture. In 2003,...

(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 {\displaystyle...

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...

William Floyd and Allen Hatcher, Incompressible surfaces in punctured-torus bundles, 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...

that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an...

q\in \mathbb {Z} } is non-zero. These are all fundamental groups of torus bundles over the circle. There are two unique geometries S o l 0 4 {\displaystyle...

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...