Homotopy class

Figure 5. Examples of Feynman paths belonging to different homotopy classes, illustrating how the winding number n is defined.

Retracing the argument used to justify point (2), it is clear that, in a multiply connected space, a given path is only coupled to those paths into which it can be continuously deformed. By definition, these are all the paths that belong to the same homotopy class. Paths belonging to different homotopy classes are thus decoupled from one another [41 5]. For a reactive system with a Cl that has the space of Fig. 1, this means that a path with a given winding number n is coupled to all paths with the same n, but is decoupled from paths with different n. As a result, the Kernel separates into [41-45]... 

Clearly, the above procedure can be continued (in principle) as many times as required. Thus, if the wave function includes n = —4 3 paths, we have simply to dehne the function I 4((t)) = —+ 8ti), and then map onto the (j) = 0 16ti cover space, which will unwind the function completely. In general, if there are h homotopy classes of Feynman paths that contribute to the Kernel, then one can unwind ihG by computing the unsymmetrised wave function ih in the 0 2hn cover space. The symmetry group of the latter will be a direct product of the symmetry group in the single space and the group... 

In this way, a complex function < )(r) can be interpreted as a map S3< S2. This is very important, since maps of this kind can be classified in homotopy classes labeled by a topological integer number called the Hopf index, so that the same topological property applies to any scalar field (provided that it is onevalued at infinity). 

Consequently, we have obtained the Cauchy data of an electromagnetic knot, a representative of the homotopy class C, for which, according to (63)... 

Introducing these vectors in (95), the expressions, for all the times, of one electromagnetic knot representative of the homotopy class C are... 

The energy, linear momentum, and angular momentum of the particular knots (114), representatives of the homotopy classes C 2, are as follows ... 

This is so because ck iv and c Fllv are generated by 4> = 5exp (ci2ns) and 0(c) = Qexp(ci2nq), respectively, which are clearly defined for any real c. Note that in this case, < ) and 0 are maps S3i >R x S1, which form only one homotopy class. Note also that the helicities vanish in this case. 

The singularities in the liquid crystals cause the deformation of the director field of liquid crystals and thus affect the symmetry of liquid crystals. This idea provides an approach to analyze the characteristics of the defects. The order vectors (or scalars, or tensors) of various liquid crystals are not the same. The director n is the order vector of the nematic liquid crystals, but the order for the cholesteric liquid crystals is a symmetric matrix, i.e., a tensor. Because the order vector space is thus a topological one, any configuration of the director field of liquid crystals is thus represented by a point in the order vector space. The order vector space (designated by M) is associated with the symmetry of liquid crystals. The topologically equivalent defects in liquid crystals constitutes the homotopy class. The complete set of homotopy classes constitutes a homotopy group, denoted Hr(M). r is the dimension of the sub-space surrounding a defect, which is related to the dimension of the defect (point, line or wall) d, and the dimension of the liquid crystal sample d by... 

A list of the most important properties of local fibrations can be found in [17]. We will only recall the following result. For simplicial sheaves denote by itlfiK, ffi) the quotient of Horrd3f), ) = So(., with respect to the equivalence relation generated by simpHcial homotopies, i.e. the set of connected compionents of the simplicial function object ), and call it the set of simplidal homotopy classes... 

The proof of Theorem 5.2.4 rests on the existence of an infinite number of hyperbolic closed geodesics. Instead of the homology group in [61] for the proof of Theorem 5.2.3 (see 2.1. above) in the case of an empty boundary dAf one should take here another topological invariant of the surface. Let F(Af) be a set of free homotopy classes in M and Il( ) II( ) homotopy classes... 

Indeed, any element of the set n( ) homotopy class of curves which... 

The equivalence classes of this equivalence relation are called homotopy classes of maps, and are denoted by [/]. 

Proposition 6.3. The composition of homotopy classes of maps is well-defined if we set... 

Let X be a CW complex. The function P from the set of homotopy classes of maps from X to 3P, [X, BF], to the set of isomorphism classes of principal P-bundles over X that takes each continuous map f X BF to the pullback of the universal bundle over BF along f is a hijection. 

A loop path within a level set F(A) is a path starting and ending at the same nuclear configuration K. Some loop paths of F(A) can be deformed continuously into one another within the level set F(A), while some others cannot. Such continuous deformations are called homotopies. Those loop paths which can be deformed into one another are said to be homotopically equivalent and to belong to the same homotopy class. 

If one is interested in the general features of reaction mechanisms, it makes sense to consider homotopy classes of paths instead of individual paths. In a strict, quantum mechanical sense, individual paths having zero width are not compatible with the Heisenberg uncertainty relation precise location orthogonal to the path implies infinite... 

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