2 |
2 |
3 \section{The blob complex} |
3 \section{The blob complex} |
4 \label{sec:blob-definition} |
4 \label{sec:blob-definition} |
5 |
5 |
6 Let $X$ be an $n$-manifold. |
6 Let $X$ be an $n$-manifold. |
7 Assume a fixed system of fields and local relations. |
7 Let $\cC$ be a fixed system of fields (enriched over Vect) and local relations. |
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8 (If $\cC$ is not enriched over Vect, we can make it so by allowing finite |
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9 linear combinations of elements of $\cC(X; c)$, for fixed $c\in \cC(\bd X)$.) |
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10 |
8 In this section we will usually suppress boundary conditions on $X$ from the notation |
11 In this section we will usually suppress boundary conditions on $X$ from the notation |
9 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
12 (e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
10 |
13 |
11 We want to replace the quotient |
14 We want to replace the quotient |
12 \[ |
15 \[ |
20 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
23 We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
21 |
24 |
22 We of course define $\bc_0(X) = \lf(X)$. |
25 We of course define $\bc_0(X) = \lf(X)$. |
23 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
26 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
24 We'll omit this sort of detail in the rest of this section.) |
27 We'll omit this sort of detail in the rest of this section.) |
25 In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
28 In other words, $\bc_0(X)$ is just the vector space of all (linearized) fields on $X$. |
26 |
29 |
27 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
30 $\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
28 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
31 Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
29 combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
32 combinations of 1-blob diagrams, where a 1-blob diagram consists of |
30 \begin{itemize} |
33 \begin{itemize} |
31 \item An embedded closed ball (``blob") $B \sub X$. |
34 \item An embedded closed ball (``blob") $B \sub X$. |
32 \item A field $r \in \cC(X \setmin B; c)$ |
35 \item A field $r \in \cC(X \setmin B; c)$ |
33 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
36 (for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
34 \item A local relation field $u \in U(B; c)$ |
37 \item A local relation field $u \in U(B; c)$ |
35 (same $c$ as previous bullet). |
38 (same $c$ as previous bullet). |
36 \end{itemize} |
39 \end{itemize} |
37 (See Figure \ref{blob1diagram}.) |
40 (See Figure \ref{blob1diagram}.) |
38 \begin{figure}[!ht]\begin{equation*} |
41 \begin{figure}[t]\begin{equation*} |
39 \mathfig{.9}{definition/single-blob} |
42 \mathfig{.9}{definition/single-blob} |
40 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
43 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} |
41 In order to get the linear structure correct, we (officially) define |
44 In order to get the linear structure correct, we (officially) define |
42 \[ |
45 \[ |
43 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
46 \bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
48 |
51 |
49 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
52 Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
50 \[ |
53 \[ |
51 (B, u, r) \mapsto u\bullet r, |
54 (B, u, r) \mapsto u\bullet r, |
52 \] |
55 \] |
53 where $u\bullet r$ denotes the linear |
56 where $u\bullet r$ denotes the field on $X$ obtained by gluing $u$ to $r$. |
54 combination of fields on $X$ obtained by gluing $u$ to $r$. |
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55 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
57 In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
56 just erasing the blob from the picture |
58 just erasing the blob from the picture |
57 (but keeping the blob label $u$). |
59 (but keeping the blob label $u$). |
58 |
60 |
59 Note that the skein space $A(X)$ |
61 Note that the skein space $A(X)$ |
60 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
62 is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
61 |
63 |
62 $\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
64 $\bc_2(X)$ is, roughly, the space of all relations (redundancies, syzygies) among the |
63 local relations encoded in $\bc_1(X)$. |
65 local relations encoded in $\bc_1(X)$. |
64 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
66 More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
65 2-blob diagrams, of which there are two types, disjoint and nested. |
67 2-blob diagrams, of which there are two types, disjoint and nested. |
66 |
68 |
67 A disjoint 2-blob diagram consists of |
69 A disjoint 2-blob diagram consists of |
70 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
72 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
71 (where $c_i \in \cC(\bd B_i)$). |
73 (where $c_i \in \cC(\bd B_i)$). |
72 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
74 \item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. \nn{We're inconsistent with the indexes -- are they 0,1 or 1,2? I'd prefer 1,2.} |
73 \end{itemize} |
75 \end{itemize} |
74 (See Figure \ref{blob2ddiagram}.) |
76 (See Figure \ref{blob2ddiagram}.) |
75 \begin{figure}[!ht]\begin{equation*} |
77 \begin{figure}[t]\begin{equation*} |
76 \mathfig{.9}{definition/disjoint-blobs} |
78 \mathfig{.9}{definition/disjoint-blobs} |
77 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
79 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} |
78 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
80 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
79 reversing the order of the blobs changes the sign. |
81 reversing the order of the blobs changes the sign. |
80 Define $\bd(B_0, B_1, u_0, u_1, r) = |
82 Define $\bd(B_0, B_1, u_0, u_1, r) = |
90 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
92 \item A field $r \in \cC(X \setmin B_0; c_0)$ |
91 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
93 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$. |
92 \item A local relation field $u_0 \in U(B_0; c_0)$. |
94 \item A local relation field $u_0 \in U(B_0; c_0)$. |
93 \end{itemize} |
95 \end{itemize} |
94 (See Figure \ref{blob2ndiagram}.) |
96 (See Figure \ref{blob2ndiagram}.) |
95 \begin{figure}[!ht]\begin{equation*} |
97 \begin{figure}[t]\begin{equation*} |
96 \mathfig{.9}{definition/nested-blobs} |
98 \mathfig{.9}{definition/nested-blobs} |
97 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
99 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} |
98 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
100 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
99 (for some $c_1 \in \cC(B_1)$) and |
101 (for some $c_1 \in \cC(B_1)$) and |
100 $r' \in \cC(X \setmin B_1; c_1)$. |
102 $r' \in \cC(X \setmin B_1; c_1)$. |
126 |
128 |
127 Now for the general case. |
129 Now for the general case. |
128 A $k$-blob diagram consists of |
130 A $k$-blob diagram consists of |
129 \begin{itemize} |
131 \begin{itemize} |
130 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
132 \item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
131 For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
133 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
132 $B_i \sub B_j$ or $B_j \sub B_i$. |
134 $B_i \sub B_j$ or $B_j \sub B_i$. |
133 (The case $B_i = B_j$ is allowed. |
135 (The case $B_i = B_j$ is allowed. |
134 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
136 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
135 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
137 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
136 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
138 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
145 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
147 \item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
146 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
148 where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
147 If $B_i = B_j$ then $u_i = u_j$. |
149 If $B_i = B_j$ then $u_i = u_j$. |
148 \end{itemize} |
150 \end{itemize} |
149 (See Figure \ref{blobkdiagram}.) |
151 (See Figure \ref{blobkdiagram}.) |
150 \begin{figure}[!ht]\begin{equation*} |
152 \begin{figure}[t]\begin{equation*} |
151 \mathfig{.9}{definition/k-blobs} |
153 \mathfig{.9}{definition/k-blobs} |
152 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
154 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
153 |
155 |
154 If two blob diagrams $D_1$ and $D_2$ |
156 If two blob diagrams $D_1$ and $D_2$ |
155 differ only by a reordering of the blobs, then we identify |
157 differ only by a reordering of the blobs, then we identify |
164 \] |
166 \] |
165 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
167 Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
166 $\overline{c}$ runs over all boundary conditions, again as described above. |
168 $\overline{c}$ runs over all boundary conditions, again as described above. |
167 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
169 $j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are splittable along all of the blobs in $\overline{B}$. |
168 |
170 |
169 The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
171 The boundary map |
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172 \[ |
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173 \bd : \bc_k(X) \to \bc_{k-1}(X) |
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174 \] |
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175 is defined as follows. |
170 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
176 Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
171 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
177 Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
172 If $B_j$ is not a twig blob, this involves only decrementing |
178 If $B_j$ is not a twig blob, this involves only decrementing |
173 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
179 the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
174 If $B_j$ is a twig blob, we have to assign new local relation labels |
180 If $B_j$ is a twig blob, we have to assign new local relation labels |
180 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
186 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
181 } |
187 } |
182 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
188 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
183 Thus we have a chain complex. |
189 Thus we have a chain complex. |
184 |
190 |
185 \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
191 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
186 |
192 but with simplices replaced by a more general class of combinatorial shapes. |
187 \nn{?? remark about dendroidal sets?; probably not} |
193 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
188 |
194 and cones, and which contains the point. |
189 |
195 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
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196 (equivalently, to each rooted tree) according to the following rules: |
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197 \begin{itemize} |
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198 \item $p(\emptyset) = pt$, where $\emptyset$ denotes a 0-blob diagram or empty tree; |
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199 \item $p(a \du b) = p(a) \times p(b)$, where $a \du b$ denotes the distant (non-overlapping) union of two blob diagrams (equivalently, join two trees at the roots); and |
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200 \item $p(\bar{b}) = \kone(p(b))$, where $\bar{b}$ is obtained from $b$ by adding an outer blob which encloses all the others. |
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201 \end{itemize} |
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202 (This correspondence works best if we thing of each twig label $u_i$ as being a difference of |
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203 two fields.) |
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204 For example, a diagram of $k$ strictly nested blobs corresponds to a $k$-simplex, while |
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205 a diagram of $k$ disjoint blobs corresponds to a $k$-cube. |
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206 |
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207 |