Compute \(H_*(S^1; E)\) and \(H^*(S^1; E)\) for \(E \to S^1\) the nontrivial bundle with fiber \(\mathbb{Z}\).
Solution
Give \(S^1\) its CW structure with one \(0\)-cell and one \(1\)-cell. For the nontrivial local system \(E\) with fiber \(\mathbb Z\), monodromy around the circle acts by \(-1\).
The cellular chain complex with local coefficients is therefore \[ 0\longrightarrow \mathbb Z \xrightarrow{(-1)-1=-2}\mathbb Z\longrightarrow0. \] Hence \[ H_1(S^1;E)=\ker(-2)=0, \qquad H_0(S^1;E)=\operatorname{coker}(-2)\cong\mathbb Z_2. \]
The cellular cochain complex is the dual two-term complex \[ 0\longrightarrow\mathbb Z \xrightarrow{-2}\mathbb Z\longrightarrow0, \] so \[ H^0(S^1;E)=0, \qquad H^1(S^1;E)\cong\mathbb Z_2. \] All other homology and cohomology groups vanish. Thus \[ \boxed{H_n(S^1;E)= \begin{cases}\mathbb Z_2,&n=0,\\0,&n\ne0, \end{cases}} \] and \[ \boxed{H^n(S^1;E)= \begin{cases}\mathbb Z_2,&n=1,\\0,&n\ne1. \end{cases}} \]