Homotopy groups

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... 

For a three-dimensional nematic liquid crystal for example, the r = 0 case corresponds for example to a defect with d = 2, which means a discli-nation wall for r = 1, d = 1 corresponds to a disclination line for r = 2, d = 0 corresponds to a disclination point. It is known that the order vector space of three dimensional nematic liquid crystals is the projection plane P2 Its homotopy group of the zero rank (r = 0) is... 

Remark 7.3.2. The above exact sequence should be compared with part of the exact sequence of homotopy groups obtained, in the theory of differentiable manifolds say, from the normal sphere bundle of SQ in /. ... 

The fundamental group ni(LDG(AT, a . Am)) of the low-density glue part of the macromolecular electron density is the one-dimensional homotopy group of the object LDG(AT, a . Am), describing the relations between equivalence classes of loops within LDG(AT, a . Am), where within each equivalence class all loops are continuously deformable into one another without any part of a loop leaving the object LDG(liT, a . Am). The product operation between loops is the continuation of one loop by another, whereas the product operation between any two equivalence classes results in an equivalence class that contains a product loop of two loops, one firom each of the two equivalence classes. 

By a similar argument, if for two proteins or for two different folding patterns Kg and Ky of the same protein the two two-dimensional homotopy groups of their respective low-density bonding contributions, VDG Kg, a , and LDG(/ft, a , Aa) are isomorphic. 

If the two homotopy groups are not isomorphic, then the group-subgroup relations among all possible LDG homotopy groups still provides a rich characterization. 

In this section we give a reformulation of the main results of [22, Ch. V] for the case of simplicial sheaves. In this context there are two noticeable differences between simplicial sheaves tmd simplicial sets. The first is that the weak homotopy type of a simplicial sheaf can not be recovered from the weak homotopy type of its Postnikoff tower unless some finitness assumptions are used (Example 1.30). The second is that a simplicial abelian group object is not necessarily weakly equivalent to the product of Eilenberg-MacLane objects corresponding to its homotopy groups (Theorem 1.34). 

Our next goal is to show that any site satisfying a fairly weak finitencss condition on cohomological dimension is a site of finite type in the sense of Definition 1.31. In order to do it we will need a description of simplicial sheaves with only one nontrivial homotopy group which is also of independent interest. 

Proposition 1.33. — Let be a simplicial sheaf which has only one nontrivial abelian homotopy group in dimension Denote by the fiber product... 

Theorem 1.34. - be a simplicial sheaf whose only nontrivial homotopy group... 

Sheaves with only one nontrivial homotopy group are related to Postnikov towers as follows. 

Proposition 1.40. — Let T be a site of finite type and U be an object ofTof cohomological dimension less than or equal to d 2. Let further JfT be a jUtrant simplicial sheaf on T which has no nontrivial homotopy groups in dimension d i.e. such that the sheaf is weakly equivalent... 

For any pointed simplicial sheaf x) and any t > 0 we get three types of presheaves of homotopy groups (or sets) ... 

Topology of Director Fields Homotopy Groups and Classification of Defects... 

The set tt (X, xq) with this operation is called the nth homotopy group of X with base point xq. 

In can be shown that when the base point is changed within the path-connected component, the new homotopy group is isomorphic to the old one. Therefore, whenever the considered topological space is path-connected, we shall skip the base point from the notation. Again, the proof is not difficult. One basically has to stretch a part of the sphere that is close to the north pole along the path connecting the old base point with the new one. 

More generally, there is a long exact sequence connecting homotopy groups of the involved spaces in any fibration. 

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