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98 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
98 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
99 Using the boundary restriction and disjoint union |
99 Using the boundary restriction and disjoint union |
100 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
100 maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
101 copies of $Y$ in $\bd X$. |
101 copies of $Y$ in $\bd X$. |
102 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
102 Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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103 (When $X$ is a disjoint union $X_1\du X_2$ the equalizer is the same as the fibered product |
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104 $\cC_k(X_1)\times_{\cC(Y)} \cC_k(X_2)$.) |
103 Then (here's the axiom/definition part) there is an injective ``gluing" map |
105 Then (here's the axiom/definition part) there is an injective ``gluing" map |
104 \[ |
106 \[ |
105 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
107 \Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
106 \] |
108 \] |
107 and this gluing map is compatible with all of the above structure (actions |
109 and this gluing map is compatible with all of the above structure (actions |