Let X be a 1-connected compact space such that the algebra H * (X; F 2) is generated by one single element. We compute the co-homology of the free loop space H * (ΛX; F 2) including the Steenrod algebra action. When X is a projective space C P n , H P n , the Cay-ley projective plane Ca P 2 or a sphere S m we obtain a splitting result for integral and mod… (More)
Let X be a space with free loop space ΛX and mod two co-homology R = H * X. We construct functors Ω λ (R) and ℓ(R) together with algebra homomorphisms e : Ω λ (R) → H * (ΛX) and ψ : ℓ(R) → H * (ES 1 × S 1 ΛX). When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms. The purpose of this paper is to present a new approach to the… (More)
Let X be a 1-connected space with free loop space ΛX. We introduce two spectral sequences converging towards H * (ΛX; Z/p) and H * ((ΛX) hT ; Z/p). The E 2-terms are certain non Abelian derived func-tors applied to H * (X; Z/p). When H * (X; Z/p) is a polynomial algebra , the spectral sequences collapse for more or less trivial reasons. If X is a sphere it… (More)
Let M be one of the projective spaces CP n , HP n for n ≥ 2 or the Cayley projective plane OP 2 , and let ΛM denote the free loop space on M. Using Morse theory methods, we prove that the suspension spectrum of (ΛM) + is homotopy equivalent to the suspension spectrum of M + wedge a family of Thom spaces of explicit vector bundles over the tangent sphere… (More)
We give a particular choice of the higher Eilenberg-MacLane maps by a recursive formula. This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras.
There is a natural evaluation map on the free loop space ΛX → X k which sends a loop to its values at the kth roots of unity. This map is equivariant with respect to the action of the cyclic group on k elements C k. We study the induced map in C k-equivariant cohomology with mod two coefficients in the cases where k = 2 m for m ≥ 1.
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We show that the space of directed paths on the $$k$$ k -skeleton of the $$n$$ n -cube is homotopy equivalent to the nerve of a certain category of flags of finite sets.
Let X be a space and write LX for its free loop space equipped with the action of the circle group T given by dilation. We compute the equivariant cohomology H * (LX hT ; Z/p) as a module over H * (BT; Z/p) when X = CP r for any positive integer r and any prime number p. The computation implies that the associated mod p Serre spectral sequence collapses… (More)
Let X be a connected space and let K = H * (X; Fp) where p is an odd prime. We construct functors ω and ℓ which approximate the co-homology of the free loop space ΛX as follows: There are morphisms