Connected sum

Connected sum

The index theorem

Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018

There are a great many consequences of both this theorem and of the Gauss-Bonnet theorem. We give three. Let M1#M2 be the connected sum of M1 and M2; this is defined by punching out disks in both manifolds and gluing along the common resulting boundaries.

Math and Art: An Introduction to Visual Mathematics, 2nd ed.,

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Journal Information

Published in Technometrics, 2022

Stan Lipovetsky

Chapter 7 of Topology introduces the basic concepts of the internal structure of topological spaces or objects which are called homotopic if one of them can be continuously deformed into the other without cutting or tearing, pasting or gluing, that is illustrated on multiple examples. Surface structures of two-manifolds are considered, with a classification theorem that every orientable two-manifold is homotopic to a sphere, a torus, or a connected sum of a finite number of tori. Torus knots are also described. It is shown that the Euler characteristic for sphere-like polyhedra equals two, while for torus it is V – E + F 0. The genus, or a maximum number of circular cuts that does not disconnect a surface, is considered. The non-orientable two- and three-manifolds are discussed, together with the Möbius bands and Klein bottles shown in various illustrations.

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The index theorem

Math and Art: An Introduction to Visual Mathematics, 2nd ed.,