1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Introduction} |
3 \section{Introduction} |
4 |
4 |
5 [Outline for intro] |
5 [some things to cover in the intro] |
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6 \begin{itemize} |
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7 \item explain relation between old and new blob complex definitions |
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8 \item overview of sections |
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9 \item state main properties of blob complex (already mostly done below) |
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10 \item give multiple motivations/viewpoints for blob complex: (1) derived cat |
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11 version of TQFT Hilbert space; (2) generalization of Hochschild homology to higher $n$-cats; |
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12 (3) ? sort-of-obvious colimit type construction; |
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13 (4) ? a generalization of $C_*(\Maps(M, T))$ to the case where $T$ is |
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14 a category rather than a manifold |
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15 \item hope to apply to Kh, contact, (other examples?) in the future |
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16 \item ?? we have resisted the temptation |
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17 (actually, it was not a temptation) to state things in the greatest |
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18 generality possible |
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19 \item related: we are being unsophisticated from a homotopy theory point of |
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20 view and using chain complexes in many places where we could be by with spaces |
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21 \item ? one of the points we make (far) below is that there is not really much |
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22 difference between (a) systems of fields and local relations and (b) $n$-cats; |
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23 thus we tend to switch between talking in terms of one or the other |
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24 \end{itemize} |
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25 |
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26 \medskip\hrule\medskip |
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27 |
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28 [Old outline for intro] |
6 \begin{itemize} |
29 \begin{itemize} |
7 \item Starting point: TQFTs via fields and local relations. |
30 \item Starting point: TQFTs via fields and local relations. |
8 This gives a satisfactory treatment for semisimple TQFTs |
31 This gives a satisfactory treatment for semisimple TQFTs |
9 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
32 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
10 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
33 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
59 |
82 |
60 \bigskip |
83 \bigskip |
61 \hrule |
84 \hrule |
62 \bigskip |
85 \bigskip |
63 |
86 |
64 We then show that blob homology enjoys the following |
87 We then show that blob homology enjoys the following properties. |
65 \ref{property:gluing} properties. |
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66 |
88 |
67 \begin{property}[Functoriality] |
89 \begin{property}[Functoriality] |
68 \label{property:functoriality}% |
90 \label{property:functoriality}% |
69 Blob homology is functorial with respect to diffeomorphisms. That is, fixing an $n$-dimensional system of fields $\cF$ and local relations $\cU$, the association |
91 Blob homology is functorial with respect to homeomorphisms. That is, |
70 \begin{equation*} |
92 for fixed $n$-category / fields $\cC$, the association |
71 X \mapsto \bc_*^{\cF,\cU}(X) |
93 \begin{equation*} |
72 \end{equation*} |
94 X \mapsto \bc_*^{\cC}(X) |
73 is a functor from $n$-manifolds and diffeomorphisms between them to chain complexes and isomorphisms between them. |
95 \end{equation*} |
74 \end{property} |
96 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
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97 \end{property} |
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98 |
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99 \nn{should probably also say something about being functorial in $\cC$} |
75 |
100 |
76 \begin{property}[Disjoint union] |
101 \begin{property}[Disjoint union] |
77 \label{property:disjoint-union} |
102 \label{property:disjoint-union} |
78 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
103 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
79 \begin{equation*} |
104 \begin{equation*} |
80 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
105 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
81 \end{equation*} |
106 \end{equation*} |
82 \end{property} |
107 \end{property} |
83 |
108 |
84 \begin{property}[A map for gluing] |
109 \begin{property}[Gluing map] |
85 \label{property:gluing-map}% |
110 \label{property:gluing-map}% |
86 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 $\bdy X_2$, |
111 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 $\bdy X_2$, |
87 there is a chain map |
112 there is a chain map |
88 \begin{equation*} |
113 \begin{equation*} |
89 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
114 \gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
90 \end{equation*} |
115 \end{equation*} |
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116 \nn{alternate version:}Given a gluing $X_\mathrm{cut} \to X_\mathrm{gl}$, there is |
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117 a natural map |
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118 \[ |
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119 \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{gl}) . |
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120 \] |
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121 (Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.) |
91 \end{property} |
122 \end{property} |
92 |
123 |
93 \begin{property}[Contractibility] |
124 \begin{property}[Contractibility] |
94 \label{property:contractibility}% |
125 \label{property:contractibility}% |
95 \todo{Err, requires a splitting?} |
126 \todo{Err, requires a splitting?} |
102 |
133 |
103 \begin{property}[Skein modules] |
134 \begin{property}[Skein modules] |
104 \label{property:skein-modules}% |
135 \label{property:skein-modules}% |
105 The $0$-th blob homology of $X$ is the usual |
136 The $0$-th blob homology of $X$ is the usual |
106 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
137 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
107 by $(\cF,\cU)$. (See \S \ref{sec:local-relations}.) |
138 by $\cC$. (See \S \ref{sec:local-relations}.) |
108 \begin{equation*} |
139 \begin{equation*} |
109 H_0(\bc_*^{\cF,\cU}(X)) \iso A^{\cF,\cU}(X) |
140 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
110 \end{equation*} |
141 \end{equation*} |
111 \end{property} |
142 \end{property} |
112 |
143 |
113 \begin{property}[Hochschild homology when $X=S^1$] |
144 \begin{property}[Hochschild homology when $X=S^1$] |
114 \label{property:hochschild}% |
145 \label{property:hochschild}% |