String cohomology groups of complex projective spaces. (English) Zbl 1134.55005
Let \(LX\) denote the free loop space on \(X := \mathbb C P^r\) (some fixed \(r\geq 1\)). Consider the space of homotopy orbits \(LX_{hS^1}:= ES^1\times_{S^1} LX\) under the canonical action of \(S^1\) on \(LX\). The purpose of this paper is to compute the equivariant singular cohomology with coefficients in the prime field \(\mathbb F_p\), \(H^*(LX_{hS^1};\mathbb F_p)\), as a module over \(H^*(BS^1;\mathbb F_p)) = \mathbb F_p[u]\). The computation uses a mixture of Morse theory and homotopy theory and does not use the fact that \(X\) is \(\mathbb F_p\)-formal and negative cyclic homology. Therefore the authors’ method extends for computations in topological \(K\)-theory or could be useful when working with non-formal manifolds. This method consists of a clever analysis of the space of geodesics on \(\mathbb CP^r\) equipped with the Fubini-Study metric and of a skillful study of the spectral sequence induced by the energy filtration.
Reviewer: Jean Claude Thomas (Angers)
MSC:
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 55P35 | Loop spaces |
| 18G50 | Nonabelian homological algebra (category-theoretic aspects) |
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