131 there is, up to homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$

   131 there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$

   142 \nn{should give a name to this property}

   142 \nn{should give a name to this property; also forward reference}

   147 \nn{now for a more detailed outline of the proof...}

   147 \nn{maybe put a little more into the outline before diving into the details.}

   148 

   148 

   149 \nn{Note: At the moment this section is very inconsistent with respect to PL versus smooth,

   150 homeomorphism versus diffeomorphism, etc.

   151 We expect that everything is true in the PL category, but at the moment our proof

   152 avails itself to smooth techniques.

   153 Furthermore, it traditional in the literature to speak of $C_*(\Diff(X))$

   154 rather than $C_*(\Homeo(X))$.}

   155 

   156 \medskip

   157 

   158 Now for the details.

   159 

   160 Notation: Let $|b| = \supp(b)$, $|p| = \supp(p)$.

   161 

   162 Choose a metric on $X$.

   163 Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero

   164 (e.g.\ $\ep_i = 2^{-i}$).

   165 Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$

   166 converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).

   167 Given a generator $p\otimes b$ of $CD_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $k$

   168 define

   169 \[

   170 	N_{i,k}(p\ot b) \deq \Nbd_{k\ep_i}(|b|) \cup \Nbd_{k\delta_i}(|p|).

   171 \]

   172 In other words, we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized

   173 by $k$), with $\ep_i$ controlling the size of the buffer around $|b|$ and $\delta_i$ controlling

   174 the size of the buffer around $|p|$.

   175 

   176 Next we define subcomplexes $G_*^{i,m} \sub CD_*(X)\otimes \bc_*(X)$.

   177 Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b)

   178 = \deg(p) + \deg(b)$.

   179 $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b)

   180 there exist codimension-zero submanifolds $V_1,\ldots,V_m \sub X$ such that each $V_j$

   181 is homeomorphic to a disjoint union of balls and

   182 \[

   183 	N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b)

   184 			\subeq V_2 \subeq \cdots \subeq V_m \subeq N_{i,k+m}(p\ot b) .

   185 \]

   186 Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$.

   187 We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.

   188 

   189 Note that $G_*^{i,m+1} \subeq G_*^{i,m}$.

   190 

   191 As sketched above and explained in detail below, 

   192 $G_*^{i,m}$ is a subcomplex where it is easy to define

   193 the evaluation map.

   194 The parameter $m$ controls the number of iterated homotopies we are able to construct.

   195 The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of

   196 $CD_*(X)\ot \bc_*(X)$.

   197 

   198 Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$.

   199 Let $p\ot b \in G_*^{i,m}$.

   200 If $\deg(p) = 0$, define

   201 \[

   202 	e(p\ot b) = p(b) ,

   203 \]

   204 where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$.

   205 For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined

   206 $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$.

   207 Choose $V_1$ as above so that 

   208 \[

   209 	N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) .

   210 \]

   211 Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V_1^j$ be the choice of neighborhood

   212 of $|p_j|\cup |b_j|$ made at the preceding stage of the induction.

   213 For all $j$, 

   214 \[

   215 	V_1^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V_1 .

   216 \]

   217 (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)

   218 We therefore have splittings

   219 \[

   220 	p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' ,

   221 \]

   222 where $p' \in CD_*(V_1)$, $p'' \in CD_*(X\setmin V_1)$, 

   223 $b' \in \bc_*(V_1)$, $b'' \in \bc_*(X\setmin V_1)$, 

   224 $e' \in \bc_*(p(V_1))$, and $e'' \in \bc_*(p(X\setmin V_1))$.

   225 (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters)

   226 near $\bd V_1)$, the expressions $p(V_1) \sub X$ and $p(X\setmin V_1) \sub X$ are

   227 unambiguous.)

   228 We also have that $\deg(b'') = 0 = \deg(p'')$.

   229 Choose $x' \in \bc_*(p(V_1))$ such that $\bd x' = f'$.

   230 This is possible by \nn{...}.

   231 Finally, define

   232 \[

   233 	e(p\ot b) \deq x' \bullet p''(b'') .

   234 \]