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

     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 $x$.