CoHochschild homology of chain coalgebras

Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex \cohoch (C) admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then \cohoch (C_{*}K) is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on \cohoch (C_{*}K) when $K$ is a simplicial suspension. The coHochschild complex construction is topologically relevant. Given two simplicial maps $g,h:K\to L$, where $K$ and $L$ are reduced, the homology of the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of $g$ and $h$, and this isomorphism respects comultiplicative structure. In particular, there a isomorphism, respecting comultiplicative structure, from the homology of \cohoch(C_{*}K) to H_{*}\op L|K|, the homology of the free loops on the geometric realization of $K$.Comment: 30 pages; some minor structural changes, new explicit formulas for comultiplicative structure in the case of suspensions; final version, to appear in JPA

Human G Protein–Coupled Receptor Gpr-9-6/Cc Chemokine Receptor 9 Is Selectively Expressed on Intestinal Homing T Lymphocytes, Mucosal Lymphocytes, and Thymocytes and Is Required for Thymus-Expressed Chemokine–Mediated Chemotaxis

TECK (thymus-expressed chemokine), a recently described CC chemokine expressed in thymus and small intestine, was found to mediate chemotaxis of human G protein–coupled receptor GPR-9-6/L1.2 transfectants. This activity was blocked by anti–GPR-9-6 monoclonal antibody (mAb) 3C3. GPR-9-6 is expressed on a subset of memory α4β7high intestinal trafficking CD4 and CD8 lymphocytes. In addition, all intestinal lamina propria and intraepithelial lymphocytes express GPR-9-6. In contrast, GPR-9-6 is not displayed on cutaneous lymphocyte antigen–positive (CLA+) memory CD4 and CD8 lymphocytes, which traffic to skin inflammatory sites, or on other systemic α4β7−CLA− memory CD4/CD8 lymphocytes. The majority of thymocytes also express GPR-9-6, but natural killer cells, monocytes, eosinophils, basophils, and neutrophils are GPR-9-6 negative. Transcripts of GPR-9-6 and TECK are present in both small intestine and thymus. Importantly, the expression profile of GPR-9-6 correlates with migration to TECK of blood T lymphocytes and thymocytes. As migration of these cells is blocked by anti–GPR-9-6 mAb 3C3, we conclude that GPR-9-6 is the principal chemokine receptor for TECK. In agreement with the nomenclature rules for chemokine receptors, we propose the designation CCR-9 for GPR-9-6. The selective expression of TECK and GPR-9-6 in thymus and small intestine implies a dual role for GPR-9-6/CCR-9, both in T cell development and the mucosal immune response

Twisting structures and morphisms up to strong homotopy

We define twisted composition products of symmetric sequences via classifying morphisms rather than twisting cochains. Our approach allows us to establish an adjunction that simultaneously generalizes a classic one for algebras and coalgebras, and the bar-cobar adjunction for quadratic operads. The comonad associated to this adjunction turns out to be, in several cases, a standard Koszul construction. The associated Kleisli categories are the "strong homotopy" morphism categories. In an appendix, we study the co-ring associated to the canonical morphism of cooperads , which is exactly the two-sided Koszul resolution of the associative operad , also known as the Alexander-Whitney co-ring

The combinatorial model for the Sullivan functor on simplicial sets

We verify the assertion made by Sullivan at the 1974 ICM congress, and previously in print, in Appendix G of the seminal paper "Differential Forms and the Topology of Manifolds" in 1973, that the rational de Rham algebra A(PL)(K) of a finite simplicial complex K has an explicit and direct combinatorial description which is closely related to that of the Stanley-Reisner face ring of K. (C) 2008 Elsevier B.V. All rights reserved