is often called a surface bundle over the circle. Mapping torus Salter, Nick; Tshishiku, Bena (21 October 2019). "Surface bundles in topology, algebraic...

mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has...

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...

In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S 1 {\displaystyle S^{1}} . Oriented circle bundles are also known as...

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

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

that of a surface bundle over the circle: such manifolds are always irreducible, and they carry a complete hyperbolic metric if and only if the monodromy...

natural structure of a fiber bundle over the circle with fiber X . {\displaystyle X.} Mapping tori of homeomorphisms of surfaces are of particular importance...

example, the Möbius band, a non-trivial line bundle over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle. A morphism...

f : X → X (see e.g. Surface bundle over the circle). Thus, HNN extensions describe the fundamental group of a self-glued space in the same way that free...

possible I-bundles over the circle S 1 {\displaystyle S^{1}} . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product...

cohomology. U ( 1 ) {\displaystyle U(1)} -principal bundles over a space M {\displaystyle M} (see circle bundle) are geometrical realizations of 1-classes in...

decomposition as a disjoint union of circles. In other words, it is a S 1 {\displaystyle S^{1}} -bundle (circle bundle) over a 2-dimensional orbifold. Many...

3-manifolds with positive first Betti number Surface bundles over the circle, this is a special case of the example above. Link complements, cf. also knot...

non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every...

are called fibered because they are surface bundles over the circle, and these manifolds are treated separately in the proof of Thurston's geometrization...

between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been...

with infinite fundamental group has a finite cover which is a surface bundle over the circle. A 3-manifold which has such a finite cover is said to virtually...

geometry and geometric topology, the Milnor–Wood inequality is an obstruction to endow circle bundles over surfaces with a flat structure. It is named...

the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the...

form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to...

surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle....

M_{f}={\frac {(I\times X)}{(1,x)\sim (0,f(x))}}} The result is a fiber bundle whose base is a circle and whose fiber is the original space X. If X is a manifold,...

not an interval bundle over a surface and M is an atoroidal then the skinning map has a fixed point. (If N is an interval bundle then the skinning map has...

and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere...

on functions on the manifold, to differential operators on the tangent bundle or frame bundle. In the case of an embedded surface, the lift to an operator...

point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid...

for the osculating circle. The curvature of curves drawn on a surface is the main tool for the defining and studying the curvature of the surface. For...

manifold Riemannian bundle metric Riemannian circle Riemannian cobordism Riemannian connection Riemannian connection on a surface Riemannian cubic Riemannian...

embedded in R4. The Klein bottle can be seen as a fiber bundle over the circle S1, with fibre S1, as follows: one takes the square (modulo the edge identifying...