1029 Define $\cl{\cC}(W)$ as a vector space via |
1029 Define $\cl{\cC}(W)$ as a vector space via |
1030 \[ |
1030 \[ |
1031 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1031 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1032 \] |
1032 \] |
1033 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1033 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
|
1034 Elements of a summand indexed by an $m$-sequences will be call $m$-simplices. |
1034 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1035 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1035 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1036 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1036 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1037 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1037 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1038 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1038 \[ |
1039 \[ |
1043 %\nn{need to say this better} |
1044 %\nn{need to say this better} |
1044 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1045 %\nn{maybe mention that there is a version that emphasizes minimal gluings (antirefinements) which |
1045 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1046 %combine only two balls at a time; for $n=1$ this version will lead to usual definition |
1046 %of $A_\infty$ category} |
1047 %of $A_\infty$ category} |
1047 |
1048 |
1048 We will call $m$ the simplex degree of the complex. |
|
1049 We can think of this construction as starting with a disjoint copy of a complex for each |
1049 We can think of this construction as starting with a disjoint copy of a complex for each |
1050 permissible decomposition (simplex degree 0). |
1050 permissible decomposition (the 0-simplices). |
1051 Then we glue these together with mapping cylinders coming from gluing maps |
1051 Then we glue these together with mapping cylinders coming from gluing maps |
1052 (simplex degree 1). |
1052 (the 1-simplices). |
1053 Then we kill the extra homology we just introduced with mapping |
1053 Then we kill the extra homology we just introduced with mapping |
1054 cylinders between the mapping cylinders (simplex degree 2), and so on. |
1054 cylinders between the mapping cylinders (the 2-simplices), and so on. |
1055 |
1055 |
1056 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1056 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. Restricting the $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1057 |
1057 |
1058 It is easy to see that |
1058 It is easy to see that |
1059 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |
1059 there are well-defined maps $\cl{\cC}(W)\to\cl{\cC}(\bd W)$, and that these maps |