58 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
58 (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
59 $Y\times F$. |
59 $Y\times F$. |
60 For simplices of dimension 1 and higher we define the map to be zero. |
60 For simplices of dimension 1 and higher we define the map to be zero. |
61 It is easy to check that this is a chain map. |
61 It is easy to check that this is a chain map. |
62 |
62 |
63 In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ |
63 In the other direction, we will define (in the next few paragraphs) |
64 and a map |
64 a subcomplex $G_*\sub \bc_*(Y\times F;\cE)$ and a map |
65 \[ |
65 \[ |
66 \phi: G_* \to \cl{\cC_F}(Y) . |
66 \phi: G_* \to \cl{\cC_F}(Y) . |
67 \] |
67 \] |
68 |
68 |
69 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
69 Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding |
78 Note that the image of $\psi$ is equal to $G_*$. |
78 Note that the image of $\psi$ is equal to $G_*$. |
79 |
79 |
80 We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. |
80 We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. |
81 Let $a$ be a generator of $G_*$. |
81 Let $a$ be a generator of $G_*$. |
82 Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$ |
82 Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$ |
83 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
83 where $b$ is a generator appearing |
84 in an iterated boundary of $a$ (this includes $a$ itself). |
84 in an iterated boundary of $a$ (this includes $a$ itself) |
|
85 and $b$ splits along $K_0\times F$. |
85 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
86 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
86 see \S\ref{ss:ncat_fields}.) |
87 see \S\ref{ss:ncat_fields}.) |
87 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
88 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
88 $b$ split according to $K_0\times F$. |
89 $b$ split according to $K_0\times F$. |
89 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
90 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
92 2-simplices which kill the homology created by the |
93 2-simplices which kill the homology created by the |
93 1-simplices, and so on. |
94 1-simplices, and so on. |
94 More formally, |
95 More formally, |
95 |
96 |
96 \begin{lemma} \label{lem:d-a-acyclic} |
97 \begin{lemma} \label{lem:d-a-acyclic} |
97 $D(a)$ is acyclic. |
98 $D(a)$ is acyclic in positive degrees. |
98 \end{lemma} |
99 \end{lemma} |
99 |
100 |
100 \begin{proof} |
101 \begin{proof} |
101 We will prove acyclicity in the first couple of degrees, and |
102 Let $P(a)$ denote the finite cone-product polyhedron composed of $a$ and its iterated boundaries. |
102 %\nn{in this draft, at least} |
103 (See Remark \ref{blobsset-remark}.) |
103 leave the general case to the reader. |
104 We can think of $D(a)$ as a cell complex equipped with an obvious |
104 |
105 map $p: D(a) \to P(a)$ which forgets the second factor. |
105 Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. |
106 For each cell $b$ of $P(a)$, let $I(b) = p\inv(b)$. |
|
107 It suffices to show that each $I(b)$ is acyclic and more generally that |
|
108 each intersection $I(b)\cap I(b')$ is acyclic. |
|
109 |
|
110 If $I(b)\cap I(b')$ is nonempty then then as a cell complex it is isomorphic to |
|
111 $(b\cap b') \times E(b, b')$, where $E(b, b')$ consists of those simplices |
|
112 $\ol{K} = (K_0,\ldots,K_l)$ such that both $b$ and $b'$ split along $K_0\times F$. |
|
113 (Here we are thinking of $b$ and $b'$ as both blob diagrams and also faces of $P(a)$.) |
|
114 So it suffices to show that $E(b, b')$ is acyclic. |
|
115 |
|
116 Let $K$ and $K'$ be two decompositions of $Y$ (i.e.\ 0-simplices) in $E(b, b')$. |
106 We want to find 1-simplices which connect $K$ and $K'$. |
117 We want to find 1-simplices which connect $K$ and $K'$. |
107 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
118 We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily |
108 the case. |
119 the case. |
109 (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) |
120 (Consider the $x$-axis and the graph of $y = e^{-1/x^2} \sin(1/x)$ in $\r^2$.) |
110 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
121 However, we {\it can} find another decomposition $L$ such that $L$ shares common |
111 refinements with both $K$ and $K'$. |
122 refinements with both $K$ and $K'$. |
|
123 This follows from Axiom \ref{axiom:vcones}, which in turn follows from the |
|
124 splitting axiom for the system of fields $\cE$. |
112 Let $KL$ and $K'L$ denote these two refinements. |
125 Let $KL$ and $K'L$ denote these two refinements. |
113 Then 1-simplices associated to the four anti-refinements |
126 Then 1-simplices associated to the four anti-refinements |
114 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
127 $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ |
115 give the desired chain connecting $(a, K)$ and $(a, K')$ |
128 give the desired chain connecting $(a, K)$ and $(a, K')$ |
116 (see Figure \ref{zzz4}). |
129 (see Figure \ref{zzz4}). |
|
130 (In the language of Axiom \ref{axiom:vcones}, this is $\vcone(K \du K')$.) |
117 |
131 |
118 \begin{figure}[t] \centering |
132 \begin{figure}[t] \centering |
119 \begin{tikzpicture} |
133 \begin{tikzpicture} |
120 \foreach \x/\label in {-3/K, 0/L, 3/K'} { |
134 \foreach \x/\label in {-3/K, 0/L, 3/K'} { |
121 \node(\label) at (\x,0) {$\label$}; |
135 \node(\label) at (\x,0) {$\label$}; |
128 \end{tikzpicture} |
142 \end{tikzpicture} |
129 \caption{Connecting $K$ and $K'$ via $L$} |
143 \caption{Connecting $K$ and $K'$ via $L$} |
130 \label{zzz4} |
144 \label{zzz4} |
131 \end{figure} |
145 \end{figure} |
132 |
146 |
133 Consider a different choice of decomposition $L'$ in place of $L$ above. |
147 Consider next a 1-cycle in $E(b, b')$, such as one arising from |
134 This leads to a cycle of 1-simplices. |
148 a different choice of decomposition $L'$ in place of $L$ above. |
135 We want to find 2-simplices which fill in this cycle. |
149 %We want to find 2-simplices which fill in this cycle. |
|
150 By Axiom \ref{axiom:vcones} we can fill in this 1-cycle with 2-simplices. |
136 Choose a decomposition $M$ which has common refinements with each of |
151 Choose a decomposition $M$ which has common refinements with each of |
137 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
152 $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. |
138 (We also require that $KLM$ antirefines to $KM$, etc.) |
153 (We also require that $KLM$ antirefines to $KM$, etc.) |
139 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
154 Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. |
140 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
155 (Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) |
173 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
188 \caption{Filling in $K$-$KL$-$L$-$K'L$-$K'$-$K'L'$-$L'$-$KL'$-$K$} |
174 \label{zzz5} |
189 \label{zzz5} |
175 \end{figure} |
190 \end{figure} |
176 |
191 |
177 Continuing in this way we see that $D(a)$ is acyclic. |
192 Continuing in this way we see that $D(a)$ is acyclic. |
|
193 By Axiom \ref{axiom:vcones} we can fill in any cycle with a V-Cone. |
178 \end{proof} |
194 \end{proof} |
179 |
195 |
180 We are now in a position to apply the method of acyclic models to get a map |
196 We are now in a position to apply the method of acyclic models to get a map |
181 $\phi:G_* \to \cl{\cC_F}(Y)$. |
197 $\phi:G_* \to \cl{\cC_F}(Y)$. |
182 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex |
198 We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex |