Hatcher 0.20

Hatcher, Algebraic Topology, Chapter 0

20. Show that the subspace X\subset\mathbb{R}^3 formed by a Klein bottle intersecting itself in a circle, as shown in Figure 1 below, is homotopy equivalent to S^1\vee S^1\vee S^2.

Hatcher 0.20
Figure 1
The space X\subset\mathbb{R}^3 described above

Proof. Let X be the figure shown above consisting of a Klein bottle intersecting itself in a circle C. The main key to constructing this result is the fact that the circle C of intersection actually yields a disk D on the “outer surface” of the Klein bottle satisfying C=\partial D. Because D is homotopy equivalent to a point, the space X can transformed to the space X_1 by way of a homotopy equivalence. The space X_1 is shown in Figure 1 below. Note that the blue and orange “loops” in Figure 1 are included to help provide some insight into the geometry present, and the “light” segments are meant to be the continuations of the “dark” ones by way of “surface transparency” of X_1. Also, the black “disc” located in the center is meant to be the point x_0 onto which D deformation retracts.

Hatcher 0.20_1
Figure 1

Before providing a pictorial reference in Figure 2, consider the following verbal description of the remaining homotopy equivalences.

Consider “stretching” the point x_0 so that a segment \ell connects it to a second point x_0'. This segment is red in Figure 2 below. This maneuver is a homotopy equivalence with inverse homotopy given by “gluing” the points x_0 and x_0' to eliminate \ell, a process that’s continuous due to the fact that \ell is homotopy equivalent to a closed interval I, and hence to a point. The result of this maneuver is the space X_2 shown in the leftmost diagram in Figure 2. Finally, consider homotopy on X_2 which glues x_0 and x_0' by moving clockwise (that is, x_0' moving “away from” x_0 instead of towards it). As mentioned previously, this is a homotopy equivalence with the homotopy inverse that “unglues and separates” x_0 and x_0'.

The result is a topological figure which is homemorphic to a sphere S^2 and which has a “wedge point” x_0=x_0' on its “surface”, along with “wedged circles” C_1\cong S^1 “outside it” (the red circle in Figure 2) and C_2\cong S^1 “inside it” (the light blue circle in Figure 2) connected via the wedge point x_0. This is sketched roughly in the second diagram of Figure 2.

Hatcher 0.20_2
Figure 2

Note that the sketches given here are extremely rough. A much clearer – and more detailed – diagram can be found here. The combination of the linked diagram with the verbal descriptions above completely characterize the series of homotopy equivalences needed to transform the original space X into the resultant space which, topologically, is homeomorphic to S^2\vee S^1\vee S^1. Hence, the result.    \square


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