A tutorial on constructions of finite complexes with specified cohomology (after Steve Mitchell and Jeff Smith)

Central to the study of modern homotopy theory is the Periodicity Theorem of Mike Hopkins and Jeff Smith, which says that any type n finite complex admits a v_n self map. Their theorem follows from the Devanitz-Hopkins-Smith Nilpotence Theorem once one has constructed at least one example of v_n self map of a type n complex. The construction of such an ur-example uses a construction due to Jeff Smith making use of the modular representation theory of the symmetric groups. This followed the first construction of a type n complex for all n by Steve Mitchell, which used the modular representation theory of the general linear groups over Z/p. The fine points of the Smith construction are not in the only published source: Ravenel's write-up in his book on the Nilpotence Theorems. I'll discuss some of this, and illustrate the ideas with a construction of a spectrum whose mod 2 cohomology is free on one generator as a module over A(3), the 1024 dimensional subalgebra of the Steenrod algebra generated by Sq^1, Sq^2, Sq^4, and Sq^8.