We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.

Theorem 1 of this note appears under a totally different point of view as Lemma 2.14 in [5]. They do not generalise to the other globally symmetric spaces of rank 1. The cohomology of the free loop space of a complex projective space is discussed as well in [4].

Let Λ(CP n ) denote the free loop space of the complex projective space CP n , i. e. the function space map(S 1 , CP n ) of unbased maps from a circle S 1 into CP n topologized with the compact open topology. In this note we show that despite the fact that the natural fibration Ω(CP n ) ֒→ Λ(CP n ) eval −→ CP n has a cross section its Serre spectral sequence does not collapse: Here eval is the evaluation map at a base point * ∈ CP n . This result was originally proven in [1] by means of the Eilenberg-Moore spectral sequence as well as in [2] by using Sullivan's theory of minimal models. Moreover it can be concluded from [3] even though it is not explicitely stated there: These results were obtained by using the energy function as a Morse function and computing the homology by determing the infinte dimensional submanifolds of the free loop space given as the critical points of the Morse function. Here we only make use of an elementary computation with the Serre spectral sequence.

In his Inventiones paper, Ziller (Invent. Math: 1-22, 1977) computed the integral homology as a graded abelian group of the free loop space of compact, globally symmetric spaces of rank 1. Chas and Sullivan (String Topology, 1999)showed that the homology of the free loop space of a compact closed orientable manifold can be equipped with a loop product and a BV-operator making it a Batalin-Vilkovisky algebra. Cohen, Jones and Yan (The loop homology algebra of spheres and projective spaces, 2004) developed a spectral sequence which converges to the loop homology as a spectral sequence of algebras. They computed the algebra structure of the loop homology of spheres and complex projective spaces by using Ziller's results and the method of Brown-Shih (Ann. of Math. 69:223-246, 1959, Publ. Math. Inst. Hautes \'Etudes Sci. 3: 93-176, 1962). In this note we compute the loop homology algebra by using only spectral sequences and the technique of universal examples. We therefore not only obtain Zillers' and Brown-Shihs' results in an elementary way, we also replace the roundabout computations of Cohen, Jones and Yan (The loop homology algebra of spheres and projective spaces, 2004) making them independent of Ziller's and Brown-Shihs' work. Moreover we offer an elementary technique which we expect can easily be generalized and applied to a wider family of spaces, not only the globally symmetric ones.Comment: 10 pages, 8 figure

We study group models for fusion systems and construct homology decompositions for the models of Robinson and Leary-Stancu type.