Homology theory : an introduction to algebraic topology by Vick J.W.

By Vick J.W.

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Correspondingly, one considers continuous maps from the n−cube to the topological space X α : In → X with the properties that the image of the boundary is one point in X α : In → X , α(s) = x0 for s ∈ ∂I n . α(I n ) is called an n−loop in X. Due to the identification of the points on the boundary these n−loops are topologically equivalent to n−spheres. e. continuous deformations of n−loops F : In × I → X and requires F (s1 , s2 , . . , 0) = α(s1 , . . , sn ) F (s1 , s2 , . . , 1) = β(s1 , . .

Definition of a coordinate system in the internal color space via the Higgs field requires φ = 0. This requirement can be enforced by the choice of form (controlled by a) and strength λ of the Higgs potential V (104). Under appropriate circumstances, quantum or thermal fluctuations will only rarely give rise to configurations where Topological Concepts in Gauge Theories 47 φ(x) vanishes at certain points and singular gauge fields (monopoles) are present. On the other hand, one expects at fixed a and λ with increasing temperature the occurrence of a phase transition to a gluon–Higgs field plasma.

In physics, one often requires differentiability of functions. In this case, the topological spaces must possess additional properties (differentiable manifolds). We start with the formal definition of homotopy. Definition: Let X, Y be smooth manifolds and f : X → Y a smooth map between them. A homotopy or deformation of the map f is a smooth map F :X ×I →Y (I = [0, 1]) with the property F (x, 0) = f (x) Each of the maps ft (x) = F (x, t) is said to be homotopic to the initial map f0 = f and the map of the whole cylinder X ×I is called a homotopy.

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