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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}. |
1523 |
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1524 It is easy to see that |
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1525 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
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1526 comprise a natural transformation of functors. |
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1527 |
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1528 |
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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. |