The minimal Sullivan model of a sphere$S^{2k+1}$ of odd dimension is the dg-algebra with a single generator $\omega_{2k+1}$ in degre $2k+1$ and vanishing differential

$d \omega_{2k+1} = 0
\,.$

The minimal Sullivan model of a sphere $S^{2k}$ of even dimension, for $k \geq 1$. is the dg-algeba with a generator $\omega_{2k}$ in degree $2k$ and another generator $\omega_{4k-1}$ in degree $4k+1$ with the differential defined by

An $n$-sphere has rational cohomology concentrated in degree $n$. Hence its minimal Sullivan model needs at least one closed generator in that degree. In the odd dimensional case one such is already sufficient, since the wedge square of that generator vanishes and hence produces no higher degree cohomology classes. But in the even degree case the wedge square $\omega_{2k}\wedge \omega_{2k}$ needs to be canceled in cohomology. That is accomplished by the second generator $\omega_{4k-1}$.