Homotopy classification

Compared with the homotopy classification, each element in the homotopy Q group corresponds to the above defects in the following way... 

Prior to such an iricntification we need some preliminary results on the homotopy classification of 2-dimensional complexes. 

One can use a derivation entirely analogous with the homotopy equivalence classes of paths and loops and the fundamental group of re on mechanisms in the nuclear configuration space M [13], leading to a group theoretical model of reaction mechanisms based on shape. The above shape equivalence of reaction paths p in space M generates a complete shape classification of all possible reaction paths for the given stoichiometry of nuclei. 

It is apparent that the two classifications are consistent, but the homotopy approach is more powerful in further analysis because there is a relation between the group elements, which is associated with the interactions between the defects. 

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

Homotopy theory is the theory of continuous deformations. The topological representation of reaction mechanisms by homotopy equivalence classes of reaction paths is based on the actual chemical equivalence of all those reaction paths that lead from some fixed reactant to some fixed product, and are "not too different" from one another. This "chemical" condition, combined with an energy constraint, corresponds to a precise topological condition two reaction paths are regarded as "not too different" if they can be continuously deformed into each other below some fixed energy value A, that is, within a level set F(A). Such paths are homotopically equivalent at energy bound A. This leads to a classification of all reaction paths a collection of all paths that are deformable into one another within F(A) is a homotopy equivalence class of paths at energy bound A. 

If there is no restriction on the value of the energy parameter A, then the above model is suitable for a classification of all reaction paths of the entire energy hypersurface into various homotopy equivalence classses. Instead of dealing with the very large set of all possible reaction paths, the analysis is reduced to the much smaller family of their equivalence classes. 

The original application of surgery to the classificatic of manifolds which are homotopy spheres due to Kervaire and Milnor HI was generalized by Barge, Lannes, t.atour and Vogel to the classification of manifolds which ate Zp-homology sphe... 

G.A.Anderson 11) developed an analogue of the Browder-W Sullivan-Wall theory (the special case P all primes in 22 ) for the classification of spaces with the P-local homotopy types of manifolds. The theory was reformulated by Taylor and Williams 12], and applied there to the classification of embeddings of manifolds in P-local homotopy spheres, the P-local version of some of the results of Browder (3). 

S-equivalence relation defined above. Farber [11,(2] has extended the classification of high-dimensional simple knots in terms of stable homotopy theory to the metastable range. 

Kojima [1] has obtained a partial classification of even-dimensional simple fibred knots in terms of Seifert formations with a homotopy pairing (the same as the one of Kearton (31). 

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