Homotopic, definition

The products of a Sharpless epoxidation, such as epoxides 12, 16, or 22, are potentially unstable in base as the anion of the alcohol can attack the epoxide 24 in the Payne rearrangement. This is easily seen with the simplest compound 12. It doesn t and we have rather given the game away by the compound numbers. The OH groups in the right hand and in the left hand compounds 12 are homotopic. Sharpless made the definitive statement of this in his propranolol synthesis.6... 

For definitions of homotopicity, enantiotopicity and diastereotopicity see Mislow and Raban (1967) and Jennings (1975). 

Proof — Let f 36- be a strict I-homotopy equivalence and be a I-homotopy inverse to f We have to show that the compositions fog and gof are equal to the corresponding identity morphisms in the I-homotopy category. By definition these compositions are I-homotopic to identity and it remains to show that two elementary I-homotopic morphisms coincide in the I-homotopy category which follows immediately from definitions. 

Consider the Euler characteristic x ) of such a surface. By definition, it is equal to 1 — rank i(Af,Z). We assume the surface M to be connected from the homotopic point of view, a surface with a non-empty boundary dM is homotopic ally equivalent to the so-called bouquet of several circles, i.e. is deduced from a finite set of circles by way of their gluing to one another at one point (Fig. 77). 

Next, we check condition (2) of Definition 7.13. First, we would like to make it crystal clear why there is anything to be checked at all. The thing is that even though we know that go/ is homotopic to idx and that go/ = id i, we cannot be sure that the homotopy between go f and idjt will fix A along the way. In fact, usually this will not happen. However, fortunately, we need to show only the existence of such a homotopy. The idea now is to start with some homotopy and then transform it to another homotopy, one that will fix A along the way. To achieve this we will need a homotopy of homotopies. 

Definition A.5. Two completely continuous vector fields having no null points on the boundary 5Kof a region V, are called homotopic if there exists a completely continuous operator H(s), depending on a parameter sg[0, 1], and having the properties... 

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