Second extra differential on odd graph complexes

Stable cohomology of graph complexes in odd dimensions

We study graph complexes related to configuration spaces and diffeomorphism groups of highly connected manifolds of odd dimension. In particular we compute the cohomology in the "high genus" limit. This paper is a continuation of previous work by Felder, Naef and the second author in which the even dimensional case is studied.

Odd two-variable Soergel bimodules and Rouquier complexes

We consider the odd analogue of the category of Soergel bimodules. In the odd case and already for two variables, the transposition bimodule cannot be merged into the generating Soergel bimodule, forcing one into a monoidal category with a larger Grothendieck ring compared to the even case. We establish biadjointness of suitable functors and develop graphical calculi in the 2-variable case for the odd Soergel category and the related singular Soergel 2-category. We describe the odd analogue of the Rouquier complexes and establish their invertibility in the homotopy category. For three variable…

Supersymmetry and cohomology of graph complexes

This is preprint HAL-00429963 (2009). I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves $\widehat{Z}_{I}\in H^{*}(\bar{\mathcal{M}}_{g,n})$ starting from the following data: an odd derivation $I$, whose square is non-zero in general, $I^{2}\neq 0$, acting on a $\mathbb{Z}/2\mathbb{Z}$-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals". Comptes Rendus Mathematique, 348, pp. 359-362, arXiv:…

从几何视角看图复形

Graph complexes from the geometric viewpoint

本文从几何视角综述图复形，介绍结合、交换和李图复形及度量图模空间，阐明后两者与胞腔链复形的对应，并修正图空间以处理结合情形。

Noncommutative differential forms and quantization of the odd symplectic category

There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for any two functions f and g. We notice that this non-commutative differential algebra has a geometrical realization as a convolution algebra of the symplectic groupoid integrating the Poisson manifold. This quantization is just a part of a quantization of the odd symplectic category (where objects are odd symplectic supermanifolds and morphisms are Lagrangian relation) in terms of Z_2-graded chain complexes. It is a straightforward consequence of the theory of BV…

A Chen model for mapping spaces and the surface product

We develop a machinery of Chen iterated integrals for higher Hochschild complexes. These are complexes whose differentials are modeled on an arbitrary simplicial set much in the same way the ordinary Hochschild differential is modeled on the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold. This is an analogue of the loop product in string topology. As an application, we show this product is homotopy invariant. We prove Hochschild-Kostant-Rosenb…