24 have the structure of a sort-of-simplicial set. |
24 have the structure of a sort-of-simplicial set. |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
25 Blob diagrams can also be equipped with a natural topology, which converts this |
26 sort-of-simplicial set into a sort-of-simplicial space. |
26 sort-of-simplicial set into a sort-of-simplicial space. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
27 Taking singular chains of this space we get $\btc_*(X)$. |
28 The details are in \S \ref{ss:alt-def}. |
28 The details are in \S \ref{ss:alt-def}. |
29 We also prove a useful lemma (\ref{small-blobs-b}) which says that we can assume that |
29 We also prove a useful result (Lemma \ref{small-blobs-b}) which says that we can assume that |
30 blobs are small with respect to any fixed open cover. |
30 blobs are small with respect to any fixed open cover. |
31 |
31 |
32 |
32 |
33 |
33 |
34 %Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
34 %Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
224 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
224 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
225 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
225 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
226 \end{itemize} |
226 \end{itemize} |
227 |
227 |
228 We can summarize the above by saying that in the typical continuous family |
228 We can summarize the above by saying that in the typical continuous family |
229 $P\to \BD_k(M)$, $p\mapsto (B_i(p), u_i(p), r(p)$, $B_i(p)$ and $r(p)$ are induced by a map |
229 $P\to \BD_k(M)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
230 $P\to \Homeo(M)$, with the twig blob labels $u_i(p)$ varying independently. |
230 $P\to \Homeo(M)$, with the twig blob labels $u_i(p)$ varying independently. |
231 We note that while have no need to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
231 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
232 if we did allow this it would not affect the truth of the claims we make below. |
232 if we did allow this it would not affect the truth of the claims we make below. |
233 In particular, we would get a homotopy equivalent complex $\btc_*(M)$. |
233 In particular, we would get a homotopy equivalent complex $\btc_*(M)$. |
234 |
234 |
235 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
235 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
236 whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |
236 whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. |