Reduced homology groups

A useful variation of the definitions above is that of reduced homology groups. To define these, let us take another look at dimension —1. Until now, we have always set C A) = 0. However, as the reader will recall from our discussion in Chapter 2, we do have an extra simplex in dimension —1, unless of course our complex is void. 

The difference between the reduced homology groups and the nonreduced ones is not large. In fact,... 

Given a 7i-space X define the reduced homology groups of X/tt with w-twisted coefficients... 

We shall write these groups as H,(X/n), the contribution of w being understood. (For w 1 these are just the usual reduced homology groups of X/ir, anyway). As C(X) is a finite-dimensional S[ji)-module chain complex the slant product chain map... 

The surface tensions for solutions of organic compounds belonging to a homologous series, for example, R(CH2)nX, show certain regularities. Roughly, Traube [145] found that for each additional CH2 group, the concentration required to give a certain surface tension was reduced by a factor of 3. This rule is manifest in Fig. lll-15b the successive curves are displaced by nearly equal intervals of 0.5 on the log C scale. 

The most general methods for the syntheses of 1,2-difunctional molecules are based on the oxidation of carbon-carbon multiple bonds (p. 117) and the opening of oxiranes by hetero atoms (p. 123fl.). There exist, however, also a few useful reactions in which an a - and a d -synthon or two r -synthons are combined. The classical polar reaction is the addition of cyanide anion to carbonyl groups, which leads to a-hydroxynitriles (cyanohydrins). It is used, for example, in Strecker s synthesis of amino acids and in the homologization of monosaccharides. The ff-hydroxy group of a nitrile can be easily substituted by various nucleophiles, the nitrile can be solvolyzed or reduced. Therefore a large variety of terminal difunctional molecules with one additional carbon atom can be made. Equally versatile are a-methylsulfinyl ketones (H.G. Hauthal, 1971 T. Durst, 1979 O. DeLucchi, 1991), which are available from acid chlorides or esters and the dimsyl anion. Carbanions of these compounds can also be used for the synthesis of 1,4-dicarbonyl compounds (p. 65f.). 

Both shortening and lengthening of the alkyl side chain result in decreased activity. While alpha-methylation of N,N-dialkyltryptamines reduces activity, alpha-methylation of the primary amines results in agents that are more active than their corresponding N,N-dimethyl counterparts. Further homologation of the alpha-methyl group decreases activity. 

Cocaine and its ethyl homolog, cocoethylene, have been assayed simultaneously using GC-NPD (Hime et al., 1991). These compounds can be derivatized by reducing the ester groups to alcohols, which can then be derivatized with a perfluoro acylating reagent to provide sensitivity for GC-ECD (Blake et al.,... 

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