2004 Fiscal Year Final Research Report Summary
Analysis of the differential (co)torsion products with algebraic models for spaces
| Project/Area Number |
14540095
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Shinshu University (2004)
Okayama University of Science (2002-2003) |
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Principal Investigator |
KURIBAYASHI Katsuhiko Shinshu University, Faculty of Science, Associate Professor, 理学部, 助教授 (40249751)
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| Project Period (FY) |
2002 – 2004
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| Keywords | (co)torsion products / algebraic models for spaces / module derivations / twisted tensor products / Eilenberg-Moore spectral sequences |
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| Research Abstract |
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
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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
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Research Products
(9 results)
