The free translation action of the circle on itself

Define $\mathrm{\Phi }([v,x])=v\cdot x$ from $ET{\times }_{T}T$ to $ET$, where $v\cdot x$ is the right Milnor action. It is well-defined because $\mathrm{\Phi }([v\cdot h,x-h])=(v\cdot h)\cdot (x-h)=v\cdot x$. Its inverse is $v\mapsto [v,0]$: $\mathrm{\Phi }([v,0])=v$, and the quotient relation gives $[v,x]=[v\cdot x,0]$. Both maps are continuous by the quotient topology and continuity of the group action, so this is a homeomorphism. By [F2], $ET$ is contractible; [F3] identifies its real singular cohomology with that of a point.

Integer translations preserve $dx$, so $-dx$ descends through the standard quotient charts to a global smooth one-form $\eta$ on the circle. The circle translations act transitively, so an invariant smooth function is constant. Every smooth one-form is $f(x)dx$ for a smooth periodic function $f$; invariance under all translations forces $f$ constant. Since higher forms vanish on a one-dimensional manifold, the invariant form algebra is spanned by $1$ and $\eta =-dx$. By [F7], the negative-exponential flow is $[x]\mapsto [x-s]$, hence $X_e=-{\mathrm{\partial }}_x$ and ${\iota }_{X_e}\eta =1$; also $d\eta =0$. Substitution in [F1] gives $d_{T}P(u)=0$ and $d_T(P(u)\eta )=-uP(u)$.

A point has one singular simplex in each dimension. If ${\sigma }_n$ is its unique $n$-simplex, then $\mathrm{\partial }{\sigma }_0=0$; for $n\ge 1$, $\mathrm{\partial }{\sigma }_n=({\sum }_{j=0}^n(-1{)}^j){\sigma }_{n-1}$, whose coefficient is $1$ for even $n$ and $0$ for odd $n$. Thus $C^n(\{\ast \};\mathbb{R})=\mathbb{R}$, and its coboundary ${\delta }^n$ is identity for odd $n$ and zero for even $n$. The cohomology quotient in [F4] is therefore $\mathbb{R}$ in degree zero and zero in every positive degree. Together with step 1.1 this proves the stated Borel cohomology.

Multiplication by $u$ is injective on $\mathbb{R}[u]$, so the differential from the odd summand $\mathbb{R}[u]\eta$ has zero kernel and image $u\mathbb{R}[u]$ in the even summand. Therefore Cartan cohomology is the quotient $\mathbb{R}[u]\mathrm{/}u\mathbb{R}[u]\cong \mathbb{R}$ in degree zero and is zero in positive degrees, matching step 2.1. The constant polynomial $1$ survives, while every positive power $u^k$ is the boundary $-d_T(u^{k-1}\eta )$; no odd class survives.