48 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
48 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
49 summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}). |
49 summarize the formal properties of the blob complex (see \S \ref{sec:properties}), describe known specializations (see \S \ref{sec:specializations}), and outline the major results of the paper (see \S \ref{sec:structure} and \S \ref{sec:applications}). |
50 %and outline anticipated future directions (see \S \ref{sec:future}). |
50 %and outline anticipated future directions (see \S \ref{sec:future}). |
51 %\nn{recheck this list after done editing intro} |
51 %\nn{recheck this list after done editing intro} |
52 |
52 |
53 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
53 The first part of the paper (sections \S \ref{sec:fields}--\S \ref{sec:evaluation}) gives the definition of the blob complex, |
54 and establishes some of its properties. |
54 and establishes some of its properties. |
55 There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is |
55 There are many alternative definitions of $n$-categories, and part of the challenge of defining the blob complex is |
56 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
56 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
57 At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex |
57 At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex |
58 associated to an $n$-manifold and an $n$-dimensional system of fields. |
58 associated to an $n$-manifold and an $n$-dimensional system of fields. |
320 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
320 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
321 |
321 |
322 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
322 \newtheorem*{thm:CH}{Theorem \ref{thm:CH}} |
323 |
323 |
324 \begin{thm:CH}[$C_*(\Homeo(-))$ action] |
324 \begin{thm:CH}[$C_*(\Homeo(-))$ action] |
325 \label{thm:evaluation}% |
|
326 There is a chain map |
325 There is a chain map |
327 \begin{equation*} |
326 \begin{equation*} |
328 e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
327 e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
329 \end{equation*} |
328 \end{equation*} |
330 such that |
329 such that |