4 \label{sec:basic-properties} |
4 \label{sec:basic-properties} |
5 |
5 |
6 In this section we complete the proofs of Properties 2-4. |
6 In this section we complete the proofs of Properties 2-4. |
7 Throughout the paper, where possible, we prove results using Properties 1-4, |
7 Throughout the paper, where possible, we prove results using Properties 1-4, |
8 rather than the actual definition of blob homology. |
8 rather than the actual definition of blob homology. |
9 This allows the possibility of future improvements to or alternatives on our definition. |
9 This allows the possibility of future improvements on or alternatives to our definition. |
10 In fact, we hope that there may be a characterisation of blob homology in |
10 In fact, we hope that there may be a characterization of the blob complex in |
11 terms of Properties 1-4, but at this point we are unaware of one. |
11 terms of Properties 1-4, but at this point we are unaware of one. |
12 |
12 |
13 Recall Property \ref{property:disjoint-union}, |
13 Recall Property \ref{property:disjoint-union}, |
14 that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
14 that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
15 |
15 |
65 |
65 |
66 \begin{proof} |
66 \begin{proof} |
67 This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. |
67 This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. |
68 \end{proof} |
68 \end{proof} |
69 |
69 |
70 Define the {\it support} of a blob diagram to be the union of all the |
70 Recall the definition of the support of a blob diagram as the union of all the |
71 blobs of the diagram. |
71 blobs of the diagram. |
72 Define the support of a linear combination of blob diagrams to be the union of the |
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73 supports of the constituent diagrams. |
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74 For future use we prove the following lemma. |
72 For future use we prove the following lemma. |
75 |
73 |
76 \begin{lemma} \label{support-shrink} |
74 \begin{lemma} \label{support-shrink} |
77 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
75 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
78 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
76 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
91 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
89 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
92 so $f$ and the identity map are homotopic. \nn{We should actually have a section with a definition of ``compatible" and this statement as a lemma} |
90 so $f$ and the identity map are homotopic. \nn{We should actually have a section with a definition of ``compatible" and this statement as a lemma} |
93 \end{proof} |
91 \end{proof} |
94 |
92 |
95 For the next proposition we will temporarily restore $n$-manifold boundary |
93 For the next proposition we will temporarily restore $n$-manifold boundary |
96 conditions to the notation. |
94 conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$. |
97 |
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98 Let $X$ be an $n$-manifold, $\bd X = Y \cup Y \cup Z$. |
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99 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
95 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
100 with boundary $Z\sgl$. |
96 with boundary $Z\sgl$. |
101 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$, |
97 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$, |
102 we have the blob complex $\bc_*(X; a, b, c)$. |
98 we have the blob complex $\bc_*(X; a, b, c)$. |
103 If $b = a$, then we can glue up blob diagrams on |
99 If $b = a$, then we can glue up blob diagrams on |
104 $X$ to get blob diagrams on $X\sgl$. |
100 $X$ to get blob diagrams on $X\sgl$. |
105 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
101 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
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102 \todo{This needs more detail, because this is false without careful attention to non-manifold components, etc.} |
106 |
103 |
107 \textbf{Property \ref{property:gluing-map}.}\emph{ |
104 \textbf{Property \ref{property:gluing-map}.}\emph{ |
108 There is a natural chain map |
105 There is a natural chain map |
109 \eq{ |
106 \eq{ |
110 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |
107 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |