104 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
104 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
105 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
105 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
106 \end{lemma} |
106 \end{lemma} |
107 |
107 |
108 The proof will be given in Section \ref{sec:localising}. |
108 The proof will be given in Section \ref{sec:localising}. |
|
109 We will actually prove the following more general result. |
|
110 Let $S$ and $T$ be an arbitrary topological spaces. |
|
111 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
|
112 %check this; or maybe just restrict the cover} |
|
113 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
|
114 from $S$ to $T$. |
|
115 Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
|
116 \nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact?} |
|
117 |
|
118 \begin{lemma} \label{extension_lemma_b} |
|
119 Let $x \in CM_k(S, T)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
|
120 Then $x$ is homotopic (rel boundary) to some $x' \in CM_k(S, T)$ which is adapted to $\cU$. |
|
121 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
|
122 If $S$ and $T$ are manifolds, the statement remains true if we replace $CM_*(S, T)$ with |
|
123 chains of smooth maps or immersions. |
|
124 \end{lemma} |
|
125 |
109 |
126 |
110 \medskip |
127 \medskip |
111 |
128 |
112 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
129 Before diving into the details, we outline our strategy for the proof of Proposition \ref{CDprop}. |
113 |
130 |