Let M be an open, connected manifold. A classical theorem of McDuff and Segal states that the sequence {Cn(M )} of configuration spaces of n unordered, distinct points in M is homologically stable with coefficients in Z -in each degree, the integral homology is eventually independent of n. The purpose of this paper is to prove that this phenomenon also holds for homology with twisted coefficients. We first define an appropriate notion of finite-degree twisted coefficient system for {Cn(M )} and then use a spectral sequence argument to deduce the result from the untwisted homological stability result of McDuff and Segal. The result and the methods are generalisations of those of Betley [Bet02] for the symmetric groups.2010 Mathematics Subject Classification: Primary 55R80; secondary 57N65.
Two mouse monoclonal antibodies against the human complement control protein, Factor H (beta 1H), are described. The antibodies are both IgG - gamma 1 - subclass and are directed against different epitopes on the human Factor H molecule. One of the antibodies, MRC OX 24, increases the cofactor activity of Factor H in Factor I-mediated cleavage of soluble C3b. The second antibody, MRC OX 23, which has no effect alone, reduces the increase in cofactor activity observed in the presence of the first antibody. However, MRC OX 24 inhibits the binding of 125I-labelled Factor H to surface-bound C3b (EAC3b). Again MRC OX 23 alone does not have an effect but decreases the inhibition in 125I-labelled Factor H binding to EAC3b observed with MRC OX 24. These studies show clearly that the interaction of Factor H with soluble C3b is different to its interaction with surface-bound C3b. In an indirect immunoprecipitation system using these monoclonal antibodies, single-chain molecules of 150 000 mol.wt. are specifically precipitated from human serum and also from the sera of other primates - rhesus monkey, cynomolgus monkey, and African green monkey. There was no precipitation from sera of cow, pig, sheep, chick, or rabbit. Using a radioimmunoassay with radiolabelled monoclonal MRC OX 23, the concentration of Factor H in human plasma was determined.
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of n-crossing diagrams for every n > 1 allows the definition of the n-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.The third author was partially supported by project KH3CF of the Initiative d'excellence at Université Sorbonne Paris Cité (and also partially by Universit Paris 13 after the suspension of the IDEX at USPC) and Universität Bonn.
The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem in [MS75] and generalize this and the homology fibration criterion of [McD75] to homology with twisted coefficients. This will be used in [MP] to identify the limiting homology of "oriented" configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.
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