Free group action

Principal Bundles and Spaces with a Free Group Action... 

In connection with their work on pseudo-free group actions Cappell and Shaneson 3) associated to a non-trivial line... 

Let G be a group functor, X a set functor. An action of G on X is a natural map G x X - X such that the individual maps G(R) x X(R)- X(R) are group actions. These will come up later for general X, but the only case of interest now is X(R)= V R, where V is a fixed k-module. If the action of G(R) here is also R-linear, we say we have a linear representation of G on V. The functor GLV(R) = Aut (F R) is a group functor a linear representation of G on V clearly assigns an automorphism to each g and is thus the same thing as a homomorphism G - GLK. If V is a finitely generated free module, then in any fixed basis automorphisms correspond to invertible matrices, and linear representations are maps to GL . 

Proposition 1.20. — Let G be an Male sheaf of groups on Sm/ and F be an etale sheaf with a free G-action. Then the folbwing conditions are equivalent ... 

Proof. It is a general fact that a finite simplicial group action on a simplicial complex is free if and only if there is no simplex whose set of vertices is preserved by the action. 

By the ketonic hydrolysis of substituted acetoacetic esters this is brought about by the action of dilute alkali in the cold, followed by acidification and boiling. The free substituted acetoacetic acid is produced, which readily undergoes decarboxylation (since it has a carboxyl and a carbonyl group on the same carbon atom) to give a ketone, for example ... 

Vulcanization. Generally this is carried out by the action of peroxides, which can cross-link the chains by abstracting hydrogen atoms from the methyl groups and allowing the resulting free radicals to couple into a cross-link. Some varieties of polysdoxanes contain some vinylmethyl siloxane units, which permit sulfur vulcanization at the double bonds. Some Hquid (short-chain) siHcones can form networks at room temperature by interaction between thek active end groups. 

A variation of the general method for the synthesis of 2-amino-selenazoles is to avoid the use of the free a-halogenocarbonyl compound and in its place react the corresponding ketone and iodine with selenourea.This procedure is also taken from thiazole chemistry. By contrast with thiourea, the reaction with selenourea needs a longer reaction time and the work up of the reaction mixture is somewhat more difficult. Usually an excess of the ketone is used. In the preparation of 2-amino-4-( n-nitrophenyl)selenazole, a very high yield, calculated on the amount of iodine used, was obtained. To explain this peculiar result, the oxidative action of the nitro group was invoked. This liberates free iodine from some of the hydrogen iodide eliminated in the condensation reaction, and the free iodine then re-enters into the reaction. 

The metabolic breakdown of triacylglycerols begins with their hydrolysis to yield glycerol plus fatty acids. The reaction is catalyzed by a lipase, whose mechanism of action is shown in Figure 29.2. The active site of the enzyme contains a catalytic triad of aspartic acid, histidine, and serine residues, which act cooperatively to provide the necessary acid and base catalysis for the individual steps. Hydrolysis is accomplished by two sequential nucleophilic acyl substitution reactions, one that covalently binds an acyl group to the side chain -OH of a serine residue on the enzyme and a second that frees the fatty acid from the enzyme. 

In subsequent studies,22 Sheehan et al. demonstrated that the action of diisopropylcarbodiimide on penicilloate 24, prepared by protection of the free primary amino group in 23 with trityl chloride (see Scheme 6b), results in the formation of the desired -lactam 25 in a very respectable yield of 67 %. In this most successful transformation, the competing azlactonization reaction is prevented by the use of a trityl group (Ph3C) to protect the C-6 amino function. Hydrogenolysis of the benzyl ester function in 25, followed by removal of the trityl protecting group with dilute aqueous HC1, furnishes 6-aminopenicillanic acid (26), a versatile intermediate for the synthesis of natural and unnatural penicillins. 

From intermediate 28, the construction of aldehyde 8 only requires a few straightforward steps. Thus, alkylation of the newly introduced C-3 secondary hydroxyl with methyl iodide, followed by hydrogenolysis of the C-5 benzyl ether, furnishes primary alcohol ( )-29. With a free primary hydroxyl group, compound ( )-29 provides a convenient opportunity for optical resolution at this stage. Indeed, separation of the equimolar mixture of diastereo-meric urethanes (carbamates) resulting from the action of (S)-(-)-a-methylbenzylisocyanate on ( )-29, followed by lithium aluminum hydride reduction of the separated urethanes, provides both enantiomers of 29 in optically active form. Oxidation of the levorotatory alcohol (-)-29 with PCC furnishes enantiomerically pure aldehyde 8 (88 % yield). 

Subjection of intermediate 16 to the action of 3 n aqueous HC1 in THF results in the formation of monocyclic lactol 14. In the presence of aqueous acid, the internal acetal grouping in intermediate 16 is hydrolyzed and lactol 14 is produced after the liberated secondary hydroxyl group attacks the terminal aldehyde carbonyl positioned five atoms away (see intermediate 15). Protection of the free aldehyde function in 14 with 1,1-dimethylhydrazine proceeds smoothly under dehydrating conditions and affords intermediate 13 in an overall yield of 72 %. 

It was projected that compound 13 could be stereoselectively linked, through its free phenolic hydroxyl group, with the anomeric carbon of intermediate 12 under suitably acidic conditions (see Scheme 8). Gratifyingly, the action of boron trifluoride etherate on a mixture of 12 and 13 in CH2CI2 at -50 °C induces a completely stereoselective glycosidation reaction, providing the desired a-ano-mer 48 in an excellent yield of 95 % from 46. It is presumed that boron trifluoride initiates cleavage of the anomeric trichloroacetimi-... 

With ring G in place, the construction of key intermediate 105 requires only a few functional group manipulations. To this end, benzylation of the free secondary hydroxyl group in 136, followed sequentially by hydroboration/oxidation and benzylation reactions, affords compound 137 in 75% overall yield. Acid-induced solvolysis of the benzylidene acetal in 137 in methanol furnishes a diol (138) the hydroxy groups of which can be easily differentiated. Although the action of 2.5 equivalents of tert-butyldimethylsilyl chloride on compound 138 produces a bis(silyl ether), it was found that the primary TBS ether can be cleaved selectively on treatment with a catalytic amount of CSA in MeOH at 0 °C. Finally, oxidation of the resulting primary alcohol using the Swem procedure furnishes key intermediate 105 (81 % yield from 138). 

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