From Geometry to Topology by H. Graham Flegg

Author:H. Graham Flegg
Language: eng
Format: epub
ISBN: 9780486138497
Publisher: Dover Publications
Published: 2012-10-12T16:00:00+00:00

it is termed a sphere with r holes.

Fig. 12.1

Fig. 12.2

Fig. 12.3

Figure 12.4 depicts a cylinder, suitably deformed, attached to a sphere with two holes so that its two end boundaries are exactly married to the boundaries of the two holes. Attaching such a cylinder in this way is termed adding a handle. A sphere with two holes to which a handle has been added is homeomorphic to a torus. The process of adding handles to pairs of holes may be continued. Any number of handles may be added to a sphere with r holes provided that for p handles there are r holes available, where r 2p. A sphere with two handles and no remaining holes is homeomorphic to a two-fold torus. Generally, a sphere with p handles and no remaining holes is homeomorphic to a p-fold torus.

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