1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \subsection{Basic properties} |
3 \subsection{Basic properties} |
4 \label{sec:basic-properties} |
4 \label{sec:basic-properties} |
5 |
5 |
6 In this section we complete the proofs of Properties 2-4. \nn{fix these numbers} |
6 In this section we complete the proofs of Properties \ref{property:disjoint-union}--\ref{property:contractibility}. |
7 Throughout the paper, where possible, we prove results using Properties 1-4, |
7 Throughout the paper, where possible, we prove results using Properties \ref{property:functoriality}--\ref{property:contractibility}, |
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 on or alternatives to 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 characterization of the blob complex 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 \ref{property:functoriality}--\ref{property:contractibility}, 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 |
16 \begin{proof}[Proof of Property \ref{property:disjoint-union}] |
16 \begin{proof}[Proof of Property \ref{property:disjoint-union}] |