17 |
17 |
18 It is also worth noting that the original idea for the blob complex came from trying |
18 It is also worth noting that the original idea for the blob complex came from trying |
19 to find a more ``local" description of the Hochschild complex. |
19 to find a more ``local" description of the Hochschild complex. |
20 |
20 |
21 Let $C$ be a *-1-category. |
21 Let $C$ be a *-1-category. |
22 Then specializing the definitions from above to the case $n=1$ we have: \nn{mention that this is dual to the way we think later} \nn{mention that this has the nice side effect of making everything splittable away from the marked points} |
22 Then specializing the definitions from above to the case $n=1$ we have: |
23 \begin{itemize} |
23 \begin{itemize} |
24 \item $\cC(pt) = \ob(C)$ . |
24 \item $\cC(pt) = \ob(C)$ . |
25 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
25 \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. |
26 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
26 Then an element of $\cC(R; c)$ is a collection of (transversely oriented) |
27 points in the interior |
27 points in the interior |
29 The intervals between the points are labeled by objects of $C$, consistent with |
29 The intervals between the points are labeled by objects of $C$, consistent with |
30 the boundary condition $c$ and the domains and ranges of the point labels. |
30 the boundary condition $c$ and the domains and ranges of the point labels. |
31 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
31 \item There is an evaluation map $e: \cC(I; a, b) \to \mor(a, b)$ given by |
32 composing the morphism labels of the points. |
32 composing the morphism labels of the points. |
33 Note that we also need the * of *-1-category here in order to make all the morphisms point |
33 Note that we also need the * of *-1-category here in order to make all the morphisms point |
34 the same way. \nn{Wouldn't it be better to just do the oriented version here? -S} |
34 the same way. |
35 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
35 \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single |
36 point (at some standard location) labeled by $x$. |
36 point (at some standard location) labeled by $x$. |
37 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
37 Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the |
38 form $y - \chi(e(y))$. |
38 form $y - \chi(e(y))$. |
39 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
39 Thus we can, if we choose, restrict the blob twig labels to things of this form. |
202 boundaries contain *, on both the right and left of *. |
202 boundaries contain *, on both the right and left of *. |
203 |
203 |
204 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
204 We claim that $J_*$ is homotopy equivalent to $\bc_*(S^1)$. |
205 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
205 Let $F_*^\ep \sub \bc_*(S^1)$ be the subcomplex where either |
206 (a) the point * is not on the boundary of any blob or |
206 (a) the point * is not on the boundary of any blob or |
207 (b) there are no labeled points or blob boundaries within distance $\ep$ of *. |
207 (b) there are no labeled points or blob boundaries within distance $\ep$ of *, |
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208 other than blob boundaries at * itself. |
208 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
209 Note that all blob diagrams are in $F_*^\ep$ for $\ep$ sufficiently small. |
209 Let $b$ be a blob diagram in $F_*^\ep$. |
210 Let $b$ be a blob diagram in $F_*^\ep$. |
210 Define $f(b)$ to be the result of moving any blob boundary points which lie on * |
211 Define $f(b)$ to be the result of moving any blob boundary points which lie on * |
211 to distance $\ep$ from *. |
212 to distance $\ep$ from *. |
212 (Move right or left so as to shrink the blob.) |
213 (Move right or left so as to shrink the blob.) |
234 Let $x \in L_*^\ep$ be a blob diagram. |
235 Let $x \in L_*^\ep$ be a blob diagram. |
235 \nn{maybe add figures illustrating $j_\ep$?} |
236 \nn{maybe add figures illustrating $j_\ep$?} |
236 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
237 If $*$ is not contained in any twig blob, we define $j_\ep(x)$ by adding |
237 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
238 $N_\ep$ as a new twig blob, with label $y - s(y)$ where $y$ is the restriction |
238 of $x$ to $N_\ep$. |
239 of $x$ to $N_\ep$. |
239 If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, \nn{I don't think we need to consider sums here} |
240 If $*$ is contained in a twig blob $B$ with label $u=\sum z_i$, |
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241 \nn{SM: I don't think we need to consider sums here} |
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242 \nn{KW: It depends on whether we allow linear combinations of fields outside of twig blobs} |
240 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
243 write $y_i$ for the restriction of $z_i$ to $N_\ep$, and let |
241 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
244 $x_i$ be equal to $x$ on $S^1 \setmin B$, equal to $z_i$ on $B \setmin N_\ep$, |
242 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
245 and have an additional blob $N_\ep$ with label $y_i - s(y_i)$. |
243 Define $j_\ep(x) = \sum x_i$. |
246 Define $j_\ep(x) = \sum x_i$. |
244 |
247 |