Seiberg—Witten—Floer Homology and Gluing Formulae

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^C structure.Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.     

Hochschild and Cyclic Homology of Ore Extensions and Some Examples of Quantum Algebras

For every Ore extension we construct a chain complex giving its Hochschild homology. As an application we compute the Hochschild and cyclic homology of an arbitrary multiparametric affine space and the Hochschild homology of the algebra of differential operators over this space, in the generic case.     

无限箭图的同调群

对任意箭图Q,我们研究路代数A=kQ的Hochschild同调群H_n(A)和同调群Tor_n^Ae(A,A),其中A^e是代数A的包络代数。在本文中,我们具体地给出了各次同调群和Hochschild同调群。     

Topological André-Quillen homology for cellular commutative S-algebras

Topological André-Quillen homology for commutative S-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative S-algebras and apply it to obtain results on minimal atomic p-local S-algebras which generalise those of Baker and May for p-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic commutative S-algebras. A. Baker was partially supported by a YFF Norwegian Research Council grant while at the University of Oslo in 2007–8, a Carnegie Trust for the Universities of Scotland grant, and Intas grants 03-51-3251 and 06-1000017-8609. H. Gilmour was supported by an EPSRC studentship. P. Reinhard was supported by an ORS grant. We would like to thank M. Basterra, P. Kropholler, M. Mandell, P. May, B. Richter, J. Rognes and S. Sagave for numerous helpful comments. We are also very grateful to the referee for encouraging us to rethink significantly issues of notation and structure, thus improving the structure of the paper.     

Phantom Maps from a Strong Homology Viewpoint

Let X and Y be connected nilpotent CW-complexes of finite type. In this paper we deal with another complete analysis of a set of phantom maps in terms of strong homology by using the derived functors. We also prove that the set of homotopy classes of maps is independent of the choice of both rational homotopy equivalent classes and completion genera.     

Homology of Bi-Grassmannian Complexes

Homology of bi-Grassmannian complex with rational coefficients is calculated. Some applications to the homological stabilization of linear groups are given.     

Homology in Abelian Lattice Models

We study Abelian lattice gauge theory defined on a simplicial complex with arbitrary topology. The use of dual objects allows one to reformulate the theory in terms of different dynamical variables; however, we avoid entirely the use of the dual cell complex. Topological modes which are present in the transformation now appear as homology classes, in contrast to the cohomology modes found in the dual cell picture. Irregularities of dual cell complexes do not arise in this approach. We treat the two and three-dimensional cases in detail, and prove a general vanishing theorem for Wilson line correlators.     

20世纪之数学

20世纪数学的重点从局部转移到整体，从低维转移到高维，从交换转移到非交换，以及从线性转移到非线性．几何学和代数学的对立统一依然存在，从哲学上反映了空间与时间之对立统一．与同调论、K-理论和李群有关的技巧和概念被广泛运用．在最近的20年间，量子场论和弦理论为数学注入新观念．理论物理学对数学的影响还将在21世纪延续下去．     

Noncommutative differential geometry on the quantum two sphere of podle?. I: An algebraic viewpoint

We compute the cyclic homology of the coordinate ringA(S _q ² (c, d)) of the quantum two sphereS _q ² (c, d) of Podle. The results are then interpreted from the point of view of Connes' noncommutative index theorem.