Solid Klein bottle

In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle.[1]

It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder \scriptstyle D^2 \times I to the bottom disk by a reflection across a diameter of the disk.

Mö x I: the circle of black points marks an absolute deformation retract of this space, and any regular neighbourhood of it has again boundary as a Klein bottle, so Mö x I is an onion of Klein bottles

Alternatively, one can visualize the solid Klein bottle as the trivial product \scriptstyle M\ddot{o}\times I, of the möbius strip and an interval \scriptstyle I=[0,1]. In this model one can see that the core central curve at 1/2 has a regular neighborhood which is again a trivial cartesian product: \scriptstyle M\ddot{o}\times[\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon] and whose boundary is a Klein bottle.

References

  1. ↑ Carter, J. Scott (1995), How Surfaces Intersect in Space: An Introduction to Topology, K & E series on knots and everything 2, World Scientific, p. 169, ISBN 9789810220662.