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596 \label{property:gluing-map}% |
596 \label{property:gluing-map}% |
597 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
597 %If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
598 %\begin{equation*} |
598 %\begin{equation*} |
599 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
599 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
600 %\end{equation*} |
600 %\end{equation*} |
601 Given a gluing $X \to X_\mathrm{gl}$, there is |
601 Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is |
602 a map |
602 a map |
603 \[ |
603 \[ |
604 \bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
604 \bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
605 \] |
605 \] |
606 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
606 natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
693 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
693 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
694 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
694 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
695 for any homeomorphic pair $X$ and $Y$, |
695 for any homeomorphic pair $X$ and $Y$, |
696 satisfying corresponding conditions. |
696 satisfying corresponding conditions. |
697 |
697 |
698 |
698 \nn{Say stuff here!} |
699 |
699 |
700 \begin{thm} |
700 \begin{thm} |
701 \label{thm:blobs-ainfty} |
701 \label{thm:blobs-ainfty} |
702 Let $\cC$ be a topological $n$-category. |
702 Let $\cC$ be a topological $n$-category. |
703 Let $Y$ be an $n{-}k$-manifold. |
703 Let $Y$ be an $n{-}k$-manifold. |