By Vick J.W.

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**Extra resources for Homology theory : an introduction to algebraic topology**

**Example text**

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.