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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 |
72 The proof will be given in Appendix \ref{sec:localising}. |
72 The proof will be given in \S\ref{sec:localising}. |
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 |