A tower of spectra that realizes a chain complex

Abstract:

This paper presents the construction of a tower of spectra ${Y_j}$ with $k$-invariants coming from the relations ${\text {S}}{{\text {q}}^1}(X{\text {S}}{{\text {q}}^{{2^{j + 1}}}}){\text {S}}{{\text {q}}^1}(X{\text {S}}{{\text {q}}^{{2^j}}}) = 0$ in $A / A{\text {S}}{{\text {q}}^1}$, for $0 \leqslant j \leqslant 5$ and $A$ = Steenrod algebra $\mod 2$, such that ${Y_5}$ has prescribed homotopy groups: ${\pi _n}({Y_5}) = Z$ (integers) if $n = {2^{j + 1}} - 2$, and zero otherwise.