Analysis of the differential (co)torsion products with algebraic models for spaces

bar和cobar型Eilenberg-Moore谱序列（EMSS）在研究分类空间、回拉空间和函数空间等许多有趣空间的上同调代数方面有很大的用处。在谱序列的构造中，微分（co）扭转积函子起着重要作用。本文从空间代数模型的角度来分析这类积函子。此外，我们试图将代数性质与空间的拓扑性质联系起来，这些性质是从考虑计算（co）扭转函子的分辨率推导出来的。结果如下：在[2]中，我们利用空间的shc-极小模型给出了EMSS的模型。并证明了谱序列的一个坍缩定理。在[3]中，我们构造了收敛于不变路径空间上同调的cobar型EMSS。设M是单连通黎曼流形。结合分析EMSS得到的事实和Tanaka关于不变测地线的结果，我们证明了M上的每一个等距都有无限多条不变测地线，如果作为代数，H^*(M; Z/2) = H^*（S^p×S^q; Z/2）且p≠q。首席研究员介绍了一个概念的模块推导与值在一个扭转乘积。在[4]中，通过推导，我们给出了对给定场的评价振动不完全非同频于零的充分条件。[5]中的一个定理断言了四维复X上的SU(n)-伴随束的同构类与束的同伦等价类重合。证明这一点的技术手段是用H^*(X; Z/p)的Hochschild同调中值的模导数。设S为不可定向曲面，BG为同调无p-扭转的单连通李群的分类空间。在[6]中，通过计算与光纤平方相关的回拉产生的EMSS，得到了上同调代数H^* (Map(S， BG)；Z/p)是明确确定的。其中Map（S， BG）表示从S到BG的所有映射的函数空间。通过[1]中关于环群分类空间的上同调的工作，统一了空间上同调的两种方法，即使用代数模型和考虑分辨率计算（co）扭转积。因此，借助于扭张量积的计算，得到了上同调H^* (BLSpin(10)；将Z/2)确定为一个模块。少

The bar and cobar type Eilenberg-Moore spectral sequences (EMSS) are of great use in studying cohomology algebras of many interesting spaces, for example, the classifying spaces, a pull-back on spaces and function spaces. In the construction of the spectral sequences, the differential (co)torsion product functors play an important role. The purpose of this research is to analyze such product functors from the viewpoint of the algebraic models for spaces. Moreover, we attempt to relate algebraic properties, which is deduced from the consideration of resolutions computing the (co)torsion functors, with topological properties of spaces.The results are as follows. In [2], we have given a model for the EMSS by applying the shc-minimal model for spaces. A collapse theorem for the spectral sequence is also proved. In [3], we have constructed the cobar type EMSS converging to the cohomology of the space of invariant paths. Let M be a simply connected Riemannian manifold. By combining the fact … More obtained by analyzing the EMSS with the result concerning invariant geodesics due to Tanaka, we have proved that every isometry on M has infinite many invariant geodesics if, as an algebra, H^*(M ; Z/2)〓H^*(S^p×S^q ; Z/2) with p≠q. The head investigator has introduced a notion of the module derivation with values in a torsion product. In [4], by using the derivation, we have given a sufficient condition for the evaluation fibration not to be totally non cohomologous to zero with respect to a given field. One of the theorems in [5] asserts that the isomorphism class of an SU(n)-adjoint bundle over 4-dimensional complex X coincides with the homotopy equivalence class of the bundle. The technical device for proving that is the module derivation with values in the Hochschild homology of H^*(X ; Z/p). Let S be a non-orientable surface and BG the classifying space of a simply connected Lie groups whose homology is p-torsion free. In [6], by calculating the EMSS which arises from a pull-back associated with a cofibre square, the cohomology algebra H^* (Map(S, BG) ; Z/p) is determined explicitly. Here Map(S, BG) denotes the function space of all maps from S to BG. The two approaches to the cohomology of spaces, namely, the use of algebraic models and the consideration of resolutions computing (co)torsion products, are unified via the work in [1] on the cohomology of the classifying space of loop groups. In consequence, with the aid of the computation of twisted tensor products, the cohomology H^* (BLSpin(10) ; Z/2) is determined as a module. Less