223 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
221 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
224 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
222 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
225 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
223 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
226 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
224 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
227 \begin{equation*} |
225 \begin{equation*} |
228 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} |
226 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
229 \end{equation*} |
227 \end{equation*} |
230 \todo{How do you write self tensor product?} |
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231 \end{itemize} |
228 \end{itemize} |
232 \end{property} |
229 \end{property} |
233 |
230 |
234 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
231 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
235 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
232 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
957 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
954 \item For any $n$-manifold $X$, with $Y \cup Y^{\text{op}}$ a codimension |
958 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
955 $0$-submanifold of its boundary, the blob homology of $X'$, obtained from |
959 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
956 $X$ by gluing along $Y$, is the $A_\infty$ self-tensor product of |
960 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
957 $\bc_*(X)$ as an $\bc_*(Y \times I)$-bimodule. |
961 \begin{equation*} |
958 \begin{equation*} |
962 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} |
959 \bc_*(X') \iso \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \!\!\!\!\!\!\xymatrix{ \ar@(ru,rd)@<-1ex>[]} |
963 \end{equation*} |
960 \end{equation*} |
964 \todo{How do you write self tensor product?} |
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965 \end{itemize} |
961 \end{itemize} |
966 |
962 |
967 Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
963 Although this gluing formula is stated in terms of $A_\infty$ categories and their (bi-)modules, it will be more natural for us to give alternative |
968 definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
964 definitions of `topological' $A_\infty$-categories and their bimodules, explain how to translate between the `algebraic' and `topological' definitions, |
969 and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |
965 and then prove the gluing formula in the topological langauge. Section \ref{sec:topological-A-infty} below explains these definitions, and establishes |