The circular disk shown in Fig. 43.14 can be deformed topologically to appear as a 2-sphere with a hole in it, as shown in Fig. 43.15. We form the real projective plane from this configuration by sewing up the hole in such a way that only diametrically opposite points on the rim of the hole are sewn together. This cannot be done in R³. Extending this idea, a 2-sphere with q holes in it, which are then sewn up by bringing together diametrically opposite points on the rims of the holes, gives a 2-sphere with q cross caps. Find the homology groups of a 2-sphere with q cross caps.

Figure 43.14

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