116 \end{itemize} |
115 \end{itemize} |
117 |
116 |
118 |
117 |
119 \section{Introduction} |
118 \section{Introduction} |
120 |
119 |
121 (motivation, summary/outline, etc.) |
120 [Outline for intro] |
122 |
121 \begin{itemize} |
123 (motivation: |
122 \item Starting point: TQFTs via fields and local relations. |
124 (1) restore exactness in pictures-mod-relations; |
123 This gives a satisfactory treatment for semisimple TQFTs |
125 (1') add relations-amongst-relations etc. to pictures-mod-relations; |
124 (i.e. TQFTs for which the cylinder 1-category associated to an |
126 (2) want answer independent of handle decomp (i.e. don't |
125 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
127 just go from coend to derived coend (e.g. Hochschild homology)); |
126 \item For non-semiemple TQFTs, this approach is less satisfactory. |
128 (3) ... |
127 Our main motivating example (though we will not develop it in this paper) |
129 ) |
128 is the $4{+}1$-dimensional TQFT associated to Khovanov homology. |
130 |
129 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
131 |
130 with a link $L \subset \bd W$. |
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131 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
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132 \item How would we go about computing $A_{Kh}(W^4, L)$? |
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133 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
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134 \nn{... $L_1, L_2, L_3$}. |
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135 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
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136 to compute $A_{Kh}(S^1\times B^3, L)$. |
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137 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
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138 corresponds to taking a coend (self tensor product) over the cylinder category |
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139 associated to $B^3$ (with appropriate boundary conditions). |
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140 The coend is not an exact functor, so the exactness of the triangle breaks. |
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141 \item The obvious solution to this problem is to replace the coend with its derived counterpart. |
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142 This presumably works fine for $S^1\times B^3$ (the answer being to Hochschild homology |
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143 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired. |
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144 If we build our manifold up via a handle decomposition, the computation |
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145 would be a sequence of derived coends. |
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146 A different handle decomposition of the same manifold would yield a different |
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147 sequence of derived coends. |
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148 To show that our definition in terms of derived coends is well-defined, we |
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149 would need to show that the above two sequences of derived coends yield the same answer. |
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150 This is probably not easy to do. |
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151 \item Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$ |
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152 which is manifestly invariant. |
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153 (That is, a definition that does not |
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154 involve choosing a decomposition of $W$. |
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155 After all, one of the virtues of our starting point --- TQFTs via field and local relations --- |
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156 is that it has just this sort of manifest invariance.) |
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157 \item The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient |
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158 \[ |
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159 \text{linear combinations of fields} \;\big/\; \text{local relations} , |
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160 \] |
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161 with an appropriately free resolution (the ``blob complex") |
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162 \[ |
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163 \cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) . |
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164 \] |
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165 Here $\bc_0$ is linear combinations of fields on $W$, |
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166 $\bc_1$ is linear combinations of local relations on $W$, |
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167 $\bc_1$ is linear combinations of relations amongst relations on $W$, |
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168 and so on. |
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169 \item None of the above ideas depend on the details of the Khovanov homology example, |
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170 so we develop the general theory in the paper and postpone specific applications |
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171 to later papers. |
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172 \item The blob complex enjoys the following nice properties \nn{...} |
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173 \end{itemize} |
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174 |
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175 \bigskip |
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176 \hrule |
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177 \bigskip |
132 |
178 |
133 We then show that blob homology enjoys the following |
179 We then show that blob homology enjoys the following |
134 \ref{property:gluing} properties. |
180 \ref{property:gluing} properties. |
135 |
181 |
136 \begin{property}[Functoriality] |
182 \begin{property}[Functoriality] |