54 |
54 |
55 \medskip |
55 \medskip |
56 |
56 |
57 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
57 If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted |
58 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
58 $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. |
|
59 For a general $k-chain$ $a\in \bc_k(X)$, define the support of $a$ to be the union |
|
60 of the supports of the blob diagrams which appear in it. |
59 |
61 |
60 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
62 If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is |
61 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
63 {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. |
62 We will sometimes abuse language and talk about ``the" support of $f$, |
64 We will sometimes abuse language and talk about ``the" support of $f$, |
63 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
65 again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that |
64 $f$ is supported on $Y$. |
66 $f$ is supported on $Y$. |
65 |
67 |
|
68 If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism |
|
69 (cf. end of \S \ref{ss:syst-o-fields}), |
|
70 we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. |
|
71 |
66 Fix $\cU$, an open cover of $X$. |
72 Fix $\cU$, an open cover of $X$. |
67 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
73 Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ |
68 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
74 of all blob diagrams in which every blob is contained in some open set of $\cU$, |
69 and moreover each field labeling a region cut out by the blobs is splittable |
75 and moreover each field labeling a region cut out by the blobs is splittable |
70 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
76 into fields on smaller regions, each of which is contained in some open set of $\cU$. |
79 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
85 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ |
80 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$. |
86 and $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$. |
81 |
87 |
82 For simplicity we will assume that all fields are splittable into small pieces, so that |
88 For simplicity we will assume that all fields are splittable into small pieces, so that |
83 $\sbc_0(X) = \bc_0$. |
89 $\sbc_0(X) = \bc_0$. |
|
90 (This is true for all of the examples presented in this paper.) |
84 Accordingly, we define $h_0 = 0$. |
91 Accordingly, we define $h_0 = 0$. |
85 |
92 |
|
93 Next we define $h_1$. |
86 Let $b\in C_1$ be a 1-blob diagram. |
94 Let $b\in C_1$ be a 1-blob diagram. |
|
95 Let $B$ be the blob of $b$. |
|
96 We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ |
|
97 and the support of $s(b)$ is contained in $B$. |
|
98 (If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition |
|
99 of $X$ where $B$ is embedded. |
|
100 See \ref{defn:configuration} and preceding discussion.) |
|
101 It then follows from \ref{disj-union-contract} that we can choose |
|
102 $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. |
|
103 |
|
104 Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series |
|
105 of small collar maps, plus a shrunken version of $b$. |
|
106 The composition of all the collar maps shrinks $B$ to a sufficiently small ball. |
|
107 |
87 Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
108 Let $\cV_1$ be an auxiliary open cover of $X$, satisfying conditions specified below. |
88 Let $B$ be the blob of $b$. |
109 Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. |
|
110 Choose a series of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ such that each has support |
|
111 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms |
|
112 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$. |
|
113 \nn{need to say this better; maybe give fig} |
|
114 Let $g_j:B\to B$ be the embedding at the $j$-th stage. |
|
115 There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ |
|
116 and $\bd c_{ij} = g_j(a_i) = g_{j-1}(a_i)$. |
|
117 Define |
|
118 \[ |
|
119 s(b) = \sum_{i,j} c_{ij} + g(b) |
|
120 \] |
|
121 and choose $h_1(b) \in \bc_1(X)$ such that |
|
122 \[ |
|
123 \bd(h_1(b)) = s(b) - b . |
|
124 \] |
|
125 |
|
126 Next we define $h_2$. |
|
127 |
89 |
128 |
90 |
129 |
91 \nn{...} |
130 \nn{...} |
92 |
131 |
93 |
|
94 |
|
95 |
|
96 |
|
97 |
|
98 %Let $k$ be the top dimension of $C_*$. |
|
99 %The construction of $h$ will involve choosing various |
|
100 |
132 |
101 |
133 |
102 |
134 |
103 |
135 |
104 \end{proof} |
136 \end{proof} |