92 \[ |
92 \[ |
93 h\bd(x) + \bd h(x) + x \in \sbc_*(X) |
93 h\bd(x) + \bd h(x) + x \in \sbc_*(X) |
94 \] |
94 \] |
95 for all $x\in C_*$. |
95 for all $x\in C_*$. |
96 |
96 |
97 For simplicity we will assume that all fields are splittable into small pieces, so that |
97 By the splittings axiom for fields, any field is splittable into small pieces. |
98 $\sbc_0(X) = \bc_0(X)$. |
98 It follows that $\sbc_0(X) = \bc_0(X)$. |
99 (This is true for all of the examples presented in this paper.) |
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100 Accordingly, we define $h_0 = 0$. |
99 Accordingly, we define $h_0 = 0$. |
101 \nn{Since we now have an axiom providing this, we should use it. (At present, the axiom is only for morphisms, not fields.)} |
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102 |
100 |
103 Next we define $h_1$. |
101 Next we define $h_1$. |
104 Let $b\in C_1$ be a 1-blob diagram. |
102 Let $b\in C_1$ be a 1-blob diagram. |
105 Let $B$ be the blob of $b$. |
103 Let $B$ be the blob of $b$. |
106 We will construct a 1-chain $s(b)\in \sbc_1(X)$ such that $\bd(s(b)) = \bd b$ |
104 We will construct a 1-chain $s(b)\in \sbc_1(X)$ such that $\bd(s(b)) = \bd b$ |
221 \begin{itemize} |
219 \begin{itemize} |
222 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
220 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
223 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
221 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
224 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
222 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
225 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
223 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
226 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{This topology is implicitly part of the data of a system of fields, but never mentioned. It should be!} |
224 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
227 \end{itemize} |
225 \end{itemize} |
228 |
226 |
229 We can summarize the above by saying that in the typical continuous family |
227 We can summarize the above by saying that in the typical continuous family |
230 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
228 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
231 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. \nn{``varying independently'' means that \emph{after} you pull back via the family of homeomorphisms to the original twig blob, you see a continuous family of labels, right? We should say this. --- Scott} |
229 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. |
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230 (``Varying independently'' means that after pulling back via the family of homeomorphisms to the original twig blob, |
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231 one sees a continuous family of labels.) |
232 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
232 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
233 if we did allow this it would not affect the truth of the claims we make below. |
233 if we did allow this it would not affect the truth of the claims we make below. |
234 In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. |
234 In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. |
235 |
235 |
236 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
236 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |