1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
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3 \section{Basic properties of the blob complex} |
3 \section{Basic properties of the blob complex} |
4 \label{sec:basic-properties} |
4 \label{sec:basic-properties} |
5 |
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6 In this section we complete the proofs of Properties 2-4. Throughout the paper, where possible, we prove results using Properties 1-4, rather than the actual definition of blob homology. This allows the possibility of future improvements to or alternatives on our definition. In fact, we hope that there may be a characterisation of blob homology in terms of Properties 1-4, but at this point we are unaware of one. |
6 In this section we complete the proofs of Properties 2-4. |
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7 Throughout the paper, where possible, we prove results using Properties 1-4, |
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8 rather than the actual definition of blob homology. |
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9 This allows the possibility of future improvements to or alternatives on our definition. |
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10 In fact, we hope that there may be a characterisation of blob homology in |
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11 terms of Properties 1-4, but at this point we are unaware of one. |
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8 Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
13 Recall Property \ref{property:disjoint-union}, |
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14 that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
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10 \begin{proof}[Proof of Property \ref{property:disjoint-union}] |
16 \begin{proof}[Proof of Property \ref{property:disjoint-union}] |
11 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
17 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
12 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
18 (putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
13 blob diagram $(b_1, b_2)$ on $X \du Y$. |
19 blob diagram $(b_1, b_2)$ on $X \du Y$. |
14 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
20 Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
15 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
21 In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
16 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
22 to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
17 a pair of blob diagrams on $X$ and $Y$. |
23 a pair of blob diagrams on $X$ and $Y$. |
18 These two maps are compatible with our sign conventions. (We follow the usual convention for tensors products of complexes, as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.) |
24 These two maps are compatible with our sign conventions. |
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25 (We follow the usual convention for tensors products of complexes, |
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26 as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.) |
19 The two maps are inverses of each other. |
27 The two maps are inverses of each other. |
20 \end{proof} |
28 \end{proof} |
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22 For the next proposition we will temporarily restore $n$-manifold boundary |
30 For the next proposition we will temporarily restore $n$-manifold boundary |
23 conditions to the notation. |
31 conditions to the notation. |
41 an $(i{+}1)$-st blob equal to all of $B^n$. |
49 an $(i{+}1)$-st blob equal to all of $B^n$. |
42 In other words, add a new outermost blob which encloses all of the others. |
50 In other words, add a new outermost blob which encloses all of the others. |
43 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
51 Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
44 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
52 the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
45 \end{proof} |
53 \end{proof} |
46 This proves Property \ref{property:contractibility} (the second half of the statement of this Property was immediate from the definitions). |
54 This proves Property \ref{property:contractibility} (the second half of the |
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55 statement of this Property was immediate from the definitions). |
47 Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
56 Note that even when there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
48 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
57 equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
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58 |
50 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
59 For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
51 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
60 where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
90 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
99 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
91 with boundary $Z\sgl$. |
100 with boundary $Z\sgl$. |
92 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
101 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
93 we have the blob complex $\bc_*(X; a, b, c)$. |
102 we have the blob complex $\bc_*(X; a, b, c)$. |
94 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
103 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
95 $X$ to get blob diagrams on $X\sgl$. This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
104 $X$ to get blob diagrams on $X\sgl$. |
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105 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
96 |
106 |
97 \textbf{Property \ref{property:gluing-map}.}\emph{ |
107 \textbf{Property \ref{property:gluing-map}.}\emph{ |
98 There is a natural chain map |
108 There is a natural chain map |
99 \eq{ |
109 \eq{ |
100 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
110 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |