67 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.

    67 Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.

    68 Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$.

    68 Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$.

    69 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.

    69 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.

    70 \end{lemma}

    70 \end{lemma}

    71 

    71 

    73 

    73 

    74 \medskip

    74 \medskip

    75 

    75 

    76 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.

    76 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CHprop}.

    77 

    77