On $H_ *(\Omega ^ {n+2}S^ {n+1};\textbf {F}_ 2)$

Abstract:

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