Manifolds of Small Dimensions
Anatoly Timofeevich Fomenko · 2019
In this chapter, we shall be guided to some extent by the material given in [34], for example. Denote by ∏ a three-dimensional manifold with boundary, bounded in R 3 by a standardly embedded two-dimensional sphere M g 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734438/ffc99044-c97a-49c7-9bd3-1988d0a4e7f6/content/inequ3_55_1.tif"/> with g handles ( Fig. 1 ). In the diagram, g = 3. For brevity, the manifold ∏ is said to be a complete pretzel of genus g. Consider two copies of complete pretzel ∏ 1 and ∏ 2 . It is well known that if we consider an arbitrary homeomorphism α:∂ ∏ 1 →∂ ∏ 2 , we can construct a three-dimensional closed compact connected and orientable manifold M ( α ). It suffices to paste together two complete pretzels ∏ 1 and ∏ 2 along the homeomorphism of their boundaries. The converse is also valid: Any three-dimensional manifold of given type can be represented as M ( α ) for a certain homeomorphism ∂:∂ ∏ 1 →∂ ∏ 2 of the surfaces of genus g. Such a representation is non-unique.