5 \input{text/top_matter.tex}

     5 \input{text/top_matter.tex}

     6 

     6 

     7 % test edit #3

     7 % test edit #3

     8 

     8 

     9 %%%%% excerpts from my include file of standard macros

     9 %%%%% excerpts from my include file of standard macros

    12 

    10 

    13 \def\z{\mathbb{Z}}

    11 \def\z{\mathbb{Z}}

    14 \def\r{\mathbb{R}}

    12 \def\r{\mathbb{R}}

    15 \def\c{\mathbb{C}}

    13 \def\c{\mathbb{C}}

    16 \def\t{\mathbb{T}}

    14 \def\t{\mathbb{T}}

   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*}

   229 \end{equation*}

   227 \end{equation*}

   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*}

   963 \end{equation*}

   960 \end{equation*}

   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