Homotopy theory

Now we briefly discuss the topological homotopy theory that classifies defects in liquid crystals and judges the stability of the defects according to the symmetry of liquid crystals. 

Similarly, homotopy theory can be applied to other liquid crystal phases, such as the cholesteric and smectic liquid crystals (Mermin, 1979 Wang, 1986). 

The first two of them play the role of two circles in the homotopy theory of schemes over S. We will use the following notations ... 

Now we go back to our A -homotopy theory of smooth scheme over a noetherian scheme of finite dimension S, in the Nisnevich topology (in fact the result which follows may hold in the more general context of site with interval). 

When S is not regular the situation becomes more complicated since Quillen s K-theory is not A -homotopy invariant on, 9 /8 and therefore the object (RilJ) B(U qBGL ) is not A -local anymore. Nevertheless, it turns out to be possible, as the following proposition shows, to describe nonnegative algebraic K-theory of any Noetherian scheme of finite dimension purely in terms of basic functoriality of the simplicial homotopy categories and the A -homotopy theory. 

V V(jKvoDSKY, The A -homotopy theory, Proceedings of the international congress of mathematicians Berlin, 1998. 33 C. Wkibel, Homotopy K-theory, Contemp. Math. 83 (1987), 461-188. Theorem... 

Again, remarkable predictions have been made on the basis of the homotopy theory regarding the properties of defects in the phase. The usual law of coalescence of defects (see 3.5.1) breaks down the combination rule is now non-Abelian. Moreover, there can be an entanglement of disclination lines, which, in turn, can give rise to what... 

The following notion is one of the most classical and important in homotopy theory. 

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