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111 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
111 %\nn{might need to restrict $S$; the proof uses partition of unity on $S$; |
112 %check this; or maybe just restrict the cover} |
112 %check this; or maybe just restrict the cover} |
113 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
113 Let $CM_*(S, T)$ denote the singular chains on the space of continuous maps |
114 from $S$ to $T$. |
114 from $S$ to $T$. |
115 Let $\cU$ be an open cover of $S$ which affords a partition of unity. |
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?} |
116 \nn{for some $S$ and $\cU$ there is no partition of unity? like if $S$ is not paracompact? |
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117 in any case, in our applications $S$ will always be a manifold} |
117 |
118 |
118 \begin{lemma} \label{extension_lemma_b} |
119 \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 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 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 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |