39 Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
39 Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct |
40 the $CH_*$ actions directly in terms of $\bc_*(X)$. |
40 the $CH_*$ actions directly in terms of $\bc_*(X)$. |
41 This was our original approach, but working out the details created a nearly unreadable mess. |
41 This was our original approach, but working out the details created a nearly unreadable mess. |
42 We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
42 We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}. |
43 |
43 |
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44 \nn{should revisit above intro after this section is done} |
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45 |
44 |
46 |
45 \subsection{Alternative definitions of the blob complex} |
47 \subsection{Alternative definitions of the blob complex} |
46 \label{ss:alt-def} |
48 \label{ss:alt-def} |
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49 |
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50 \newcommand\sbc{\bc^{\cU}} |
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51 |
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52 In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs) |
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53 of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$. |
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54 |
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55 \medskip |
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56 |
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57 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
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58 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
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59 |
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60 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
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61 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
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62 We will sometimes abuse language and talk about ``the" support of $f$, |
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63 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
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64 $f$ is supported on $Y$. |
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65 |
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66 Fix $\cU$, an open cover of $X$. |
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67 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
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68 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
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69 and moreover each field labeling a region cut out by the blobs is splittable |
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70 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
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71 |
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72 \begin{thm}[Small blobs] \label{thm:small-blobs-xx} |
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73 The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. |
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74 \end{thm} |
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75 |
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76 \begin{proof} |
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77 It suffices to show that for any finitely generated pair of subcomplexes |
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78 $(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))$ |
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79 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
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80 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$. |
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81 |
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82 For simplicity we will assume that all fields are splittable into small pieces, so that |
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83 $\sbc_0(X) = \bc_0$. |
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84 Accordingly, we define $h_0 = 0$. |
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85 |
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86 Let $b\in C_1$ be a 1-blob diagram. |
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87 Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
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88 Let $B$ be the blob of $b$. |
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89 |
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90 |
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91 \nn{...} |
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92 |
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93 |
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94 |
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95 |
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96 |
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97 |
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98 %Let $k$ be the top dimension of $C_*$. |
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99 %The construction of $h$ will involve choosing various |
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100 |
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101 |
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102 |
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103 |
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104 \end{proof} |
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105 |
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106 |
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107 |
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108 |
47 |
109 |
48 |
110 |
49 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
111 \subsection{Action of \texorpdfstring{$\CH{X}$}{C_*(Homeo(M))}} |
50 \label{ss:emap-def} |
112 \label{ss:emap-def} |
51 |
113 |