1518 

  1518 

  1519 \medskip

  1519 \medskip

  1520 

  1520 

  1521 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. 

  1521 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. 

  1522 Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.

  1522 Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}.

  1529 

  1523 

  1530 \begin{lem}

  1524 \begin{lem}

  1531 \label{lem:colim-injective}

  1525 \label{lem:colim-injective}

  1532 Let $W$ be a manifold of dimension less than $n$.  Then for each

  1526 Let $W$ be a manifold of dimension less than $n$.  Then for each

  1533 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective.

  1527 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective.