On H(Ωn+2Sn+1; F2)

In this paper, we study H*Ωn+2Sn+1. Here Ω X denotes the space of pointed maps S1→ X, and H*represents homology modulo 2. We show that the Eilenberg-Moore spectral sequence$\operatorname{Tor}^{\ast\ast}_{H^\ast\Omega^{n+1}_0S^{n+1}} (F_2, F_2) \Rightarrow H^\ast\Omega^{n+2}S^{n+1}$collapses, and we identify the kernel of the Whitehead product map Ωn+1p*: H*Ωn+3S2n+1→ H*Ωn+1Sn. These observations yield two different descriptions of H*Ωn+2Sn+1up to extension.