We prove that ΩSn(2), Sn{2r}, and Ω2 Sn(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 Ω2 Sn, 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.
Homology, Homotopy and Applications, Vol. 9 (2007), No. 1, pp.1-43.