Cohomology of the free loop space of a complex projective space

Abstract

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.

Loop homology of spheres and complex projective spaces

“…From the E 2 -page it is not obvious that d 2 (y) = 0, but, again the knowledge of the cohomolgy Serre spectral sequence implies d 2 (y) = 0. Quoting [15] again we obtain with the Universal Coefficient Theorem that d 2n (z) = (n + 1)x n ⊗ y.…”

“…Our own method goes as follows. For even dimensional spheres recall that completely analogously to [15] there is map of fibrations: For the term E 2 of the loop homology spectral sequence we also have the following diagram. The next observation is crucial.…”

“…Our own method goes as follows. For even dimensional spheres recall that completely analogously to [15] there is map of fibrations:…”

“…In [18] Smith computed the characteristic 0 cohomology of the free loop space for a complex projective space by making use of the Eilenberg Moore spectral sequence, the Eilenberg Moore theorem and constructing an explicit Koszul-type resolution. In [15] we only made use of an elementary computation with the Serre spectral sequence. A related result of Smith in charasteristic p appeared in [19].…”

The free loop space homology of $$(n-1)$$ ( n - 1 ) -connected 2n-manifolds

“…Unfortunately, this theorem is generally not true for fields with nonzero characteristic [20]. Beyond these results, the BV-algebra over various coefficient rings has been completely determined for spheres [10,20,25], certain Stiefel manifolds [24], Lie groups [17], and projective spaces [10,16,22,27,28], using a mixture of techniques ranging from homotopy theoretic to geometric, as well as the well-known connections to Hochschild cohomology.…”