Decompositions of polyhedral products for shifted complexes

The conjecture of Bahri, Bendersky, Cohen and Gitler (2010)  [3] on wedge decompositions of polyhedral products of shifted complexes is settled affirmatively. As a corollary, it is proved that the homotopy type of the complement of a coordinate subspace arrangement associated with a shifted complex, tensored with R^r${\mathbb{R}}^r$ for any r≥1$r\ge 1$, has the homotopy type of a wedge of spheres.     

Hom complexes and hypergraph colorings

Babson and Kozlov (2006) [2] studied Hom-complexes of graphs with a focus on graph colorings. In this paper, we generalize Hom-complexes to r-uniform hypergraphs (with multiplicities) and study them mainly in connection with hypergraph colorings. We reinterpret a result of Alon, Frankl and Lovász (1986) [1] by Hom-complexes and show a hierarchy of known lower bounds for the chromatic numbers of r-uniform hypergraphs (with multiplicities) using Hom-complexes.     

Diastereoselective tandem Michael-intramolecular Wittig reactions of a cyclic phosphonium ylide with 8-phenylmenthyl enoates.

The diastereoselective tandem Michael-intramolecular Wittig reactions of a five-membered cyclic phosphonium ylide 2 using 8-phenylmenthyl enoates were examined. The reaction of the phosphonium ylide with 8-phenylmenthyl cinnamate followed by the hydrolysis of the resulting enol ether 4a afforded (3R,4S)-4-(diphenylphosphinyl)-3-phenylcycloheptanone (3R,4S)-5a as the major isomer. The diastereoselectivity of the initial tandem reactions was estimated to be 94:6 from the 31P NMR of a mixture of the diastereomeric ketal derivatives 6a and 6'a which were obtained by the reaction of 5a with (2R,3R)-2,3-butanediol, and the absolute configuration of the major isomer was determined by the single-crystal X-ray analysis. Similar reactions using some 8-phenylmenthyl alkenoates were attempted. As a result, it was clarified that the corresponding trans-ketones 5b-d were obtained and that the diastereomer ratios of their ketal derivatives were 60:40-73:27.     

The Space of Loops on a Homogeneous Space and Rational Equivalence

We shall construct a map which is a rational equivalence with applications to phantom maps,spaces of the same n-type and genus sets. This answers in theaffirmative a question of C. McGibbon. 1991 Mathematics SubjectClassification 55Q05.     

On the Gray index conjecture for phantom maps

We study the Gray index, a numerical invariant for phantom maps. It has been conjectured that the only phantom map between finite-type spaces with infinite Gray index is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of the phantom maps are restricted to being simply connected finite complexes.As a result of the counter example, we can show that SNT^∞(X) can be non-trivial for some space X of finite type.     

Reactions of cyclic sulfur ylides with some carbonyl compounds

The reaction of a six‐membered sulfonium ylide 5 with aldehydes or ketones afforded the oxirane derivatives 6a–d as a mixture of cis and trans isomers in excellent yields. In addition, the same reactions, using five‐ or six‐membered cyclic oxosulfonium ylides 7 and 11 , gave the corresponding oxirane derivatives in good yields. Moreover, the reaction of 11 with two equimolar amounts of base and 4‐hexen‐3‐one afforded the cyclooctene oxide derivative 16 with high stereoselectivity in 59% yield via a sequential Michael–Michael‐type addition of the ylide and the resulting enolate ion followed by an intramolecular Corey–Chaykovsky reaction. © 2002 Wiley Periodicals, Inc. Heteroatom Chem 13:216–222, 2002; Published online in Wiley Interscience (www.interscience.wiley.com). DOI 10.1002/hc.10022     

Stable indecomposability of loop spaces on symplectic groups

We prove that is stably indecomposable if or .

     

Preparation of new nitrogen-bridged heterocycles. 55. Reinvestigation of the bromination/dehydrobromination of ethyl 3-[2-(methylthio)indolizin-3-yl]acrylate derivatives

The bromination/dehydrobromination reactions of ethyl 3-[1-alkoxycarbonyl-2-(methylthio)indolizin-3-yl]acrylates were reinvestigated. Reactions of the title compounds with two equivalents of bromine, followed by heating of the resulting reaction mixture and then treatment with a base gave the unexpected dialkyl 7-bromothieno[2,3-b]indolizine-2,9-dicarboxylates, while similar reactions using benzyltrimethylammonium tribromide as a brominating agent afforded only non-brominated thieno[2,3-b]indolizine derivatives, which were converted to the corresponding 7-bromo derivatives upon further treatment with bromine.