Properties of Anick’s spaces

Abstract:

We prove three useful properties of Anick’s space $T^{2n-1}(p^{r})$. First, at odd primes a map from $P^{2n}(p^{r})$ into a homotopy commutative, homotopy associative $H$-space $X$ can be extended to a unique $H$-map from $T^{2n-1}(p^{r})$ into $X$. Second, at primes larger than $3$, $T^{2n-1}(p^{r})$ is itself homotopy commutative and homotopy associative. And third, the first two properties combine to show that the order of the identity map on $T^{2n-1}(p^{r})$ is $p^{r}$.