Higher order cohomology operations and minimal atomicity

Abstract

We prove that $\Omega S^n_{(2)}, S^n\{2^r\}$, and $\Omega^2 S^n_{(2)}$,are minimal atomic spaces for appropriate values of $n$. We do this by using secondary and tertiary cohomology operations to prove that, above the Hurewicz dimension, no elements in the mod 2 homology of the cited spaces are in the image of the Hurewicz homomorphism. In the case of ­$\Omega^2 S^n$, we construct and exploit an appropriate filtration to facilitate the use of higher order cohomology operations. An appendix consisting of an examination of the coefficients in Adams’ factorization is included. 1.