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

BAR和Cobar型Eilenberg-Moore谱序列(EMSS)在许多有趣的空间的上同调代数的研究中有很大的用处，例如分类空间、空间和函数空间上的拉回。在谱序列的构造中，微分(余)扭积函子起着重要的作用。本研究的目的是从空间的代数模型的角度来分析这类乘积函子。此外，我们还尝试将计算(余)挠函子的分辨率所得的代数性质与空间的拓扑性质联系起来，结果如下。在文[2]中，我们应用空间的SHC-极小模型给出了EMSS的一个模型。还证明了谱序列的一个崩塌定理。在文[3]中，我们构造了收敛于不变路空间的上同调的Cobar型EMSS。设M是单连通的黎曼流形。通过结合…的事实进一步利用田中关于不变测地线的结果分析了M上的EMSS，证明了M上的每条等距线都有无穷多条不变测地线，如果H^*(M；Z/2)〓H^*(S^p×S^q；Z/2)有p≠q。在文[4]中，我们利用这个导数，给出了赋值纤维关于给定域不完全非上同调为零的一个充分条件。文[5]中的一个定理断言：4维复X上的SU(N)-伴随丛的同构类与该丛的同伦等价类重合。证明这一点的技术手段是用H^*(X；Z/p)的Hochschild同调中值的模导出。设S是不可定向曲面，BG是同调p-挠自由的单连通李群的分类空间。在文[6]中，通过计算与上纤维方相关的拉回产生的EMSS，显式地确定了上同调代数H^*(Map(S，BG)；Z/p)。这里的映射(S，BG)表示从S到BG的所有映射的函数空间。通过文献[1]中关于循环群分类空间上同调的工作，统一了空间上同调的两种方法，即利用代数模型和考虑分解计算(余)扭积。因此，借助于扭曲张量积的计算，上同调H^*(BLSpin(10)；Z/2)被确定为模。较少

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