Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 24 / 6 / i / Solution

Use cohomological degree, and write the Sullivan minimal models asandThe subscripts indicate degrees. The even generators generate a polynomial ring and the odd generators an exterior algebra. Their cohomology rings are respectively $\mathbb{Q}[x]/(x^{n+1})$ and $\mathbb{Q}[u]/(u^2)$, since each odd differential kills exactly the indicated non-zero-divisor polynomial relation. These are the usual projective-space and even-sphere minimal models.

Orient the top classes so that the degree-one condition is $f^{\ast }[u]=[x^n]$. A representative model map must send $u$ to $x^n$. It must send $v$ to an element of degree $4n-1$ with differential $x^{2n}$. The element $x^{n-1}y$ has exactly these properties. ThusThe differential check is $d(x^{n-1}y)=x^{n-1}{x}^{n+1}=x^{2n}=(f^{\ast }u{)}^2$. In these degrees there is no other available monomial: even powers of $x$ have even degree, and an odd monomial is a power of $x$ times $y$. Hence the coefficient is forced by the differential. For $n=1$ the two models coincide and the map is the identity model map.