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1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
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 |
5 |
6 In this section we complete the proofs of Properties 1-4. 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 1-5. Throughout the paper, where possible, we prove results using Properties 1-5, 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 1-5, but at this point we are unaware of one. |
7 |
7 |
8 Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
8 Recall Property \ref{property:disjoint-union}, that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
9 |
9 |
10 \begin{proof}[Proof of Property \ref{property:disjoint-union}] |
10 \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 |
11 Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
104 `Natural' means natural with respect to the actions of diffeomorphisms. |
104 `Natural' means natural with respect to the actions of diffeomorphisms. |
105 } |
105 } |
106 |
106 |
107 This map is very far from being an isomorphism, even on homology. |
107 This map is very far from being an isomorphism, even on homology. |
108 We fix this deficit in Section \ref{sec:gluing} below. |
108 We fix this deficit in Section \ref{sec:gluing} below. |
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109 |
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110 As we pointed out earlier, Property \ref{property:skein-modules} is immediate from the definitions. |