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In this paper, we study      H ∗  Ω  n + 2 S n + 1 {H_\ast }{\Omega ^{n + 2}}{S^{n + 1}}       . Here      Ω  X \Omega X       denotes the space of pointed maps      S 1 →  X {S^1} \to X      , and      H ∗  {H_\ast }       represents homology modulo      2 2      . We show that the Eilenberg-Moore spectral sequence      Tor H ∗  Ω  0 n + 1 S n + 1 ∗  ∗  ⁡  ( F 2 , F 2 ) ⇒  H ∗  Ω  n + 2 S n + 1 \operatorname {Tor}_{{H^\ast }\Omega _0^{n + 1}{S^{n + 1}}}^{\ast \ast }({F_{2,}}{F_2}) \Rightarrow {H^\ast }{\Omega ^{n + 2}}{S^{n + 1}}       collapses, and we identify the kernel of the Whitehead product map      Ω  n + 1 p ∗  : H ∗  Ω  n + 3 S 2 n + 1 →  H ∗  Ω  n + 1 S n {\Omega ^{n + 1}}{p_\ast }:{H_\ast }{\Omega ^{n + 3}}{S^{2n + 1}} \to {H_\ast }{\Omega ^{n + 1}}{S^n}       . These observations yield two different descriptions of      H ∗  Ω  n + 2 S n + 1 {H_\ast }{\Omega ^{n + 2}}{S^{n + 1}}       up to extension.

On 𝐻_{*}(Ωⁿ⁺²𝑆ⁿ⁺¹;𝐹₂)