The purpose of this paper is to consider the space \(\Omega^k S^{2n+1}\langle 2n+1\rangle_{(p)}\), the \(k\)-fold loops on the \(2n+1\)-connected cover of the \(2n+1\) sphere localized at a prime \(p > 3\). An H-space \(X\) has an H-space exponent \(M\) if the power map \(X \to X\), \(x\mapsto x^M\) is null homotopic. The H-space has mod \(p\) exponent if \(M\) is an H-space exponent for \(X\) localized at \(p\). In the author’s thesis, he established that \(S^3\) has mod \(p\) exponent \(p\) for \(p\) an odd prime. Based on this work, Cohen, Moore and Neisendorfer established \(p^n\) as the best possible mod \(p\) exponent for \(S^{2n+1}\) for \(p\geq 5\) (later extended to \(p\geq 3\) by Neisendorfer). The author and Neisendorfer showed that \(\Omega^k S^{2n+1}\langle 2n+1\rangle_{(p)}\) has no mod \(p\) H-space exponent for \(k < 2n-1\). Cohen, Moore, and Neisendorfer showed that \(p^n\) is an H-space exponent for \(k \geq 2n\). This leaves the case \(k = 2n-1\) open and lying between the two outcomes. In this paper the author shows that \(\Omega^{2n-1} S^{2n+1}\langle 2n+1\rangle_{(p)}\) has H-space exponent \(p^n\) for \(p \geq 5\). The negative results of Neisendorfer and the author are also given another proof. The main techniques are the properties of the various (localized) spaces associated to the study of the double suspension and the \(p\)th power map. If \(X \to X\) is the \(p\)th power map, \(x \mapsto x^p\), then denote its fiber by \(X\{p\}\). A basic tool in the work of Cohen, Moore, and Neisendorfer is the splitting \(\Omega^2 S^{2p+1}\{p\} \simeq \Omega^2 S^3\langle 3\rangle \times C(p)\) where \(C(n)\) is the fiber of the double suspension \(E^2\: S^{2n-1} \to \Omega^2 S^{2p+1}\). Anick has introduced spaces \(T^{2n-1}_\infty\{p\}\) that fit into homotopy fibrations \[ C(n) \to T^{2n-1}_\infty\{p\} \to \Omega S^{2n+1}\{p\} \] with the projection an H-map. The main theorem establishes \(T^{2p-1}_\infty\{p\} \simeq \Omega S^3\langle 3\rangle\) and that \(\Omega S^{2p+1}\{p\}\simeq \Omega S^3 \langle 3\rangle \times B(p)\) where \(B(p)\) is a delooping of \(C(p)\). From earlier work \(\Omega S^3\langle 3\rangle\) has H-space exponent \(p\) from which it follows that \(\Omega^{2n-1} S^{2n+1}\langle 2n+1\rangle\) has H-space exponent \(p^n\). The new proof of the Neisendorfer-Selick negative results is based on Miller’s Theorem (resolving the Sullivan conjecture).
For the entire collection see [Zbl 0890.00047].