40 Throughout, we have resisted the temptation to work in the greatest possible generality. |
40 Throughout, we have resisted the temptation to work in the greatest possible generality. |
41 %(Don't worry, it wasn't that hard.) |
41 %(Don't worry, it wasn't that hard.) |
42 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category |
42 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category |
43 with sufficient limits and colimits would do. |
43 with sufficient limits and colimits would do. |
44 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
44 We could also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories). |
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45 |
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46 {\bf Note:} For simplicity, we will assume that all manifolds are unoriented and piecewise linear, unless stated otherwise. |
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47 In fact, all the results in this paper also hold for smooth manifolds, |
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48 as well as manifolds equipped with an orientation, spin structure, or $\mathrm{Pin}_\pm$ structure. |
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49 We will use ``homeomorphism" as a shorthand for ``piecewise linear homeomorphism". |
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50 The reader could also interpret ``homeomorphism" to mean an isomorphism in whatever category of manifolds we happen to |
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51 be working in (e.g.\ spin piecewise linear, oriented smooth, etc.). |
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52 In the smooth case there are additional technical details concerning corners and gluing |
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53 which we have omitted, since |
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54 most of the examples we are interested in require only a piecewise linear structure. |
45 |
55 |
46 |
56 |
47 \subsection{Structure of the paper} |
57 \subsection{Structure of the paper} |
48 The subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
58 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}), |
59 summarize the formal properties of the blob complex (see \S \ref{sec:properties}), |