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1 %!TEX root = ../../blob1.tex |
1 %!TEX root = ../../blob1.tex |
2 |
2 |
3 \section{Families of Diffeomorphisms} \label{sec:localising} |
3 \section{Families of Diffeomorphisms} \label{sec:localising} |
4 |
4 |
5 In this appendix we provide the proof of |
5 In this appendix we provide the proof of |
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6 \nn{should change this to the more general \ref{extension_lemma_b}} |
6 |
7 |
7 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
8 \begin{lem*}[Restatement of Lemma \ref{extension_lemma}] |
8 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
9 Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. |
9 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
10 Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. |
10 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |
11 Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. |