nLab
Möbius strip

Contents

Idea

graphics grabbed from Lawson 03

The Möbius strip is the topological space obtained from the “open strip”, hence the square, by gluing two of its opposite sites but after applying half of a full rotation to one of them

Regarded as a manifold, then the Möbius strip is among the simplest examples of manifolds which is not orientable. Regarding as a real vector bundle over the circle, then the Möbius strip is among the simplest examples of a non-trivial vector bundle.

Realizations

As a topological space, the Möbius strip is

  • the quotient topological space obtained from the square [0,1] 2[0,1]^2 by the equivalence relation which identifies two of the opposite sides, but with opposite orientation

    ((x 1,y 1)(x 2,y 2))((x 1,y 1)=(x 2,y 2)or(x 1=(1x 2){0,1}andy 1=(1y 2))) ((x_1,y_1) \sim (x_2,y_2)) \;\;\Leftrightarrow \;\; \left( (x_1,y_1) = (x_2, y_2) \;\text{or}\; \left( x_1 = (1-x_2) \in \{0,1\} \;\text{and}\; y_1 = (1-y_2) \right) \right)

For the realization of the Möbius strip as a topological vector bundle (a real line bundle) see there, this example.

References

Named after August Möbius.

  • Terry Lawson, Topology: A Geometric Approach, Oxford University Press (2003) (pdf)

Revised on June 13, 2017 08:34:23 by Urs Schreiber (131.220.184.222)