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In mathematics, a Möbius strip, Möbius band, or Möbius loop [a] is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE .
The Möbius strip is a nontrivial bundle over the circle. Perhaps the simplest example of a nontrivial bundle E {\displaystyle E} is the Möbius strip . It has the circle that runs lengthwise along the center of the strip as a base B {\displaystyle B} and a line segment for the fiber F {\displaystyle F} , so the Möbius strip is a bundle of the ...
This is an annulus, not a Möbius strip. It has two half-twists and is thus orientable. The standard Möbius strip has the unknot for a boundary but is not a Seifert surface for the unknot because it is not orientable. The "checkerboard" coloring of the usual minimal crossing projection of the trefoil knot gives a Mobius strip with three half ...
The homology of a topological space X is a set of topological invariants of X represented by its homology groups (), (), (), … where the homology group () describes, informally, the number of holes in X with a k-dimensional boundary.
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
Notice that there is an induced action of the mapping class group on the homology (and cohomology) of the space X. This is because (co)homology is functorial and Homeo 0 acts trivially (because all elements are isotopic, hence homotopic to the identity, which acts trivially, and action on (co)homology is invariant under homotopy).
The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S 1. When glued trivially (with g UV =1) one obtains the trivial bundle, but with the non-trivial gluing of g UV =1 on one overlap and g UV =-1 on the second overlap, one obtains the non-trivial bundle E, the Möbius strip
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]