582 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
582 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
583 |
583 |
584 \begin{example}[Blob complexes of balls (with a fiber)] |
584 \begin{example}[Blob complexes of balls (with a fiber)] |
585 \rm |
585 \rm |
586 \label{ex:blob-complexes-of-balls} |
586 \label{ex:blob-complexes-of-balls} |
587 Fix an $m$-dimensional manifold $F$ and system of fields $\cE$. |
587 Fix an $n-k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
588 We will define an $A_\infty$ $(n-m)$-category $\cC$. |
588 We will define an $A_\infty$ $k$-category $\cC$. |
589 When $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = \cE(X\times F)$. |
589 When $X$ is a $m$-ball or $m$-sphere, with $m<k$, define $\cC(X) = \cE(X\times F)$. |
590 When $X$ is an $(n-m)$-ball, |
590 When $X$ is an $k$-ball, |
591 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
591 define $\cC(X; c) = \bc^\cE_*(X\times F; c)$ |
592 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
592 where $\bc^\cE_*$ denotes the blob complex based on $\cE$. |
593 \end{example} |
593 \end{example} |
594 |
594 |
595 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
595 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
1092 %\nn{start with (less general) tensor products; maybe change this later} |
1092 %\nn{start with (less general) tensor products; maybe change this later} |
1093 |
1093 |
1094 |
1094 |
1095 |
1095 |
1096 |
1096 |
1097 \subsection{Morphisms of $A_\infty$ 1-cat modules} |
1097 \subsection{Morphisms of $A_\infty$ $1$-category modules} |
1098 \label{ss:module-morphisms} |
1098 \label{ss:module-morphisms} |
1099 |
1099 |
1100 In order to state and prove our version of the higher dimensional Deligne conjecture |
1100 In order to state and prove our version of the higher dimensional Deligne conjecture |
1101 (Section \ref{sec:deligne}), |
1101 (Section \ref{sec:deligne}), |
1102 we need to define morphisms of $A_\infty$ 1-category modules and establish |
1102 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1103 some of their elementary properties. |
1103 some of their elementary properties. |
1104 |
1104 |
1105 To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1105 To motivate the definitions which follow, consider algebras $A$ and $B$, right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1106 \begin{eqnarray*} |
1106 \begin{eqnarray*} |
1107 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1107 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1149 We will denote an element of the summand indexed by $\olD$ by |
1149 We will denote an element of the summand indexed by $\olD$ by |
1150 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
1150 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
1151 The boundary map is given by |
1151 The boundary map is given by |
1152 \begin{align*} |
1152 \begin{align*} |
1153 \bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
1153 \bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
1154 & \qquad + (-1)^l \olD\ot\bd(m\ot\cbar\ot n) |
1154 & \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ |
|
1155 & \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n |
1155 \end{align*} |
1156 \end{align*} |
1156 where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
1157 where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
1157 boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
1158 boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
1158 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
1159 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
1159 |
1160 |
1161 \[ |
1162 \[ |
1162 \prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
1163 \prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
1163 \] |
1164 \] |
1164 where $(\psi(D_0)[l])^*$ denotes the linear dual. |
1165 where $(\psi(D_0)[l])^*$ denotes the linear dual. |
1165 The boundary is given by |
1166 The boundary is given by |
1166 \begin{eqnarray*} |
1167 \begin{align} |
1167 (\bd f)(\olD\ot m\ot\cbar\ot n) &=& f(\olD\ot\bd(m\ot\cbar)\ot n) + |
1168 \label{eq:tensor-product-boundary} |
1168 f(\olD\ot m\ot\cbar\ot \bd n) + \\ |
1169 (-1)^{\deg f +1} (\bd f)(\olD\ot m\ot\cbar\ot n) & = f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) + f((\bd_+ \olD)\ot m\ot\cbar\ot n) + \\ |
1169 & & \;\; f((\bd_+ \olD)\ot m\ot\cbar\ot n) + f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) . |
1170 & \qquad + (-1)^{l} f(\olD\ot\bd m\ot\cbar \ot n) + (-1)^{l + \deg m} f(\olD\ot m\ot\bd \cbar \ot n) + \notag \\ |
1170 \end{eqnarray*} |
1171 & \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
1171 (Again, we are ignoring signs.) \nn{put signs in} |
1172 \end{align} |
1172 |
1173 |
1173 Next we define the dual module $(_\cC\cN)^*$. |
1174 Next we define the dual module $(_\cC\cN)^*$. |
1174 This will depend on a choice of interval $J$, just as the tensor product did. |
1175 This will depend on a choice of interval $J$, just as the tensor product did. |
1175 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
1176 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
1176 to chain complexes. |
1177 to chain complexes. |
1186 |
1187 |
1187 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1188 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1188 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1189 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1189 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1190 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1190 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
1191 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
1191 Recall that $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
1192 Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
1192 Then for each such $\olD$ we have a degree $l$ map |
1193 Then for each such $\olD$ we have a degree $l$ map |
1193 \begin{eqnarray*} |
1194 \begin{eqnarray*} |
1194 \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
1195 \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
1195 m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
1196 m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
1196 \end{eqnarray*} |
1197 \end{eqnarray*} |
1230 \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
1231 \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
1231 \] |
1232 \] |
1232 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals |
1233 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals |
1233 which are dropped off the right side. |
1234 which are dropped off the right side. |
1234 (Either $\cbar'$ or $\cbar''$ might be empty.) |
1235 (Either $\cbar'$ or $\cbar''$ might be empty.) |
1235 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ \nn{give ref?}, |
1236 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
|
1237 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1236 we have |
1238 we have |
1237 \begin{eqnarray*} |
1239 \begin{eqnarray*} |
1238 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1240 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1239 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') . |
1241 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') . |
1240 \end{eqnarray*} |
1242 \end{eqnarray*} |
|
1243 \nn{put in signs, rearrange terms to match order in previous formulas} |
1241 Here $\gl$ denotes the module action in $\cY_\cC$. |
1244 Here $\gl$ denotes the module action in $\cY_\cC$. |
1242 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1245 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1243 |
1246 |
1244 Note that if $\bd g = 0$, then each |
1247 Note that if $\bd g = 0$, then each |
1245 \[ |
1248 \[ |
1536 \begin{align*} |
1539 \begin{align*} |
1537 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
1540 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)), |
1538 \end{align*} |
1541 \end{align*} |
1539 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
1542 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
1540 |
1543 |
1541 We now give two motivating examples, as theorems constructing other homological systems of fields, |
|
1542 |
|
1543 |
|
1544 \begin{thm} |
|
1545 For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as |
|
1546 \begin{equation*} |
|
1547 \Xi(M) = \CM{M}{X}. |
|
1548 \end{equation*} |
|
1549 \end{thm} |
|
1550 |
|
1551 \begin{thm} |
|
1552 Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by |
|
1553 \begin{equation*} |
|
1554 \cF^{\times F}(M) = \cB_*(M \times F, \cF). |
|
1555 \end{equation*} |
|
1556 \end{thm} |
|
1557 We might suggestively write $\cF^{\times F}$ as $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories. |
|
1558 |
|
1559 |
|
1560 In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields. |
|
1561 |
|
1562 |
|
1563 \begin{thm} |
|
1564 \begin{equation*} |
|
1565 \cB_*(M, \Xi) \iso \Xi(M) |
|
1566 \end{equation*} |
|
1567 \end{thm} |
|
1568 |
|
1569 \begin{thm}[Product formula] |
|
1570 Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields, |
|
1571 there is a quasi-isomorphism |
|
1572 \begin{align*} |
|
1573 \cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) |
|
1574 \end{align*} |
|
1575 \end{thm} |
|
1576 |
|
1577 \begin{question} |
|
1578 Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
|
1579 \end{question} |
|
1580 |
|
1581 \hrule |
1544 \hrule |