Computation of H *(Λ(M))

We have seen in the previous chapters that the space $$\mathbb{F}_k (M)$$ can be described as a twisted product of simpler spaces when M is ℝn+1 or Sn+1. The simpler spaces are bouquets of n-dimensional spheres when M = ℝn+1; when M = Sn+1, they include the Stiefel manifold O _n+2,2 of orthonormal 2-frames in ℝn+2, as well. We have also seen that the space $$\Omega \mathbb{F}_k (M)$$ of based loops splits as a product of the loop spaces of the split factors as spaces, but not as loop spaces. A natural question to ask is whether the space of free loops $$\Lambda \mathbb{F}_k (M)$$ splits, at the homology level, as a tensor product of the homology of the split factors of $$\mathbb{F}_k (M)$$. We shall see in this chapter that this is the exception: it is true for k = 3, but not in general.