equal
deleted
inserted
replaced
72 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
72 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
73 to be the union of the blobs of $b$. |
73 to be the union of the blobs of $b$. |
74 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
74 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
75 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
75 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
76 |
76 |
|
77 %%%%% the following is true in spirit, but technically incorrect if blobs are not embedded; |
|
78 %%%%% we only use this once, so move lemma and proof to Hochschild section |
|
79 \noop{ %%%%%%%%%% begin \noop |
77 For future use we prove the following lemma. |
80 For future use we prove the following lemma. |
78 |
81 |
79 \begin{lemma} \label{support-shrink} |
82 \begin{lemma} \label{support-shrink} |
80 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
83 Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
81 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
84 subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
92 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
95 Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
93 Note that if a diagram $b'$ is part of $\bd b$ then $T(b') \sub T(b)$. |
96 Note that if a diagram $b'$ is part of $\bd b$ then $T(b') \sub T(b)$. |
94 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}), |
97 Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models, \S\ref{sec:moam}), |
95 so $f$ and the identity map are homotopic. |
98 so $f$ and the identity map are homotopic. |
96 \end{proof} |
99 \end{proof} |
|
100 } %%%%%%%%%%%%% end \noop |
97 |
101 |
98 For the next proposition we will temporarily restore $n$-manifold boundary |
102 For the next proposition we will temporarily restore $n$-manifold boundary |
99 conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$. |
103 conditions to the notation. Let $X$ be an $n$-manifold, with $\bd X = Y \cup Y \cup Z$. |
100 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
104 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
101 with boundary $Z\sgl$. |
105 with boundary $Z\sgl$. |