diff -r 62cd552bf8c9 -r c3aace2330ac text/evmap.tex --- a/text/evmap.tex Sat Jul 04 06:48:22 2009 +0000 +++ b/text/evmap.tex Sat Jul 04 18:44:35 2009 +0000 @@ -177,11 +177,11 @@ Let $p\ot b$ be a generator of $CD_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b) = \deg(p) + \deg(b)$. $p\ot b$ is (by definition) in $G_*^{i,m}$ if either (a) $\deg(p) = 0$ or (b) -there exist codimension-zero submanifolds $V_1,\ldots,V_m \sub X$ such that each $V_j$ +there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$ is homeomorphic to a disjoint union of balls and \[ - N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) - \subeq V_2 \subeq \cdots \subeq V_m \subeq N_{i,k+m}(p\ot b) . + N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b) + \subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) . \] Further, we require (inductively) that $\bd(p\ot b) \in G_*^{i,m}$. We also require that $b$ is splitable (transverse) along the boundary of each $V_l$. @@ -191,9 +191,10 @@ As sketched above and explained in detail below, $G_*^{i,m}$ is a subcomplex where it is easy to define the evaluation map. -The parameter $m$ controls the number of iterated homotopies we are able to construct. +The parameter $m$ controls the number of iterated homotopies we are able to construct +(Lemma \ref{mhtyLemma}). The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of -$CD_*(X)\ot \bc_*(X)$. +$CD_*(X)\ot \bc_*(X)$ (Lemma \ref{xxxlemma}). Next we define a chain map (dependent on some choices) $e: G_*^{i,m} \to \bc_*(X)$. Let $p\ot b \in G_*^{i,m}$. @@ -204,38 +205,119 @@ where $p(b)$ denotes the obvious action of the diffeomorphism(s) $p$ on the blob diagram $b$. For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined $e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$. -Choose $V_1$ as above so that +Choose $V = V_0$ as above so that \[ - N_{i,k}(p\ot b) \subeq V_1 \subeq N_{i,k+1}(p\ot b) . + N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) . \] -Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V_1^j$ be the choice of neighborhood +Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood of $|p_j|\cup |b_j|$ made at the preceding stage of the induction. For all $j$, \[ - V_1^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V_1 . + V^j \subeq N_{i,(k-1)+1}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V . \] (The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.) We therefore have splittings \[ p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' , \] -where $p' \in CD_*(V_1)$, $p'' \in CD_*(X\setmin V_1)$, -$b' \in \bc_*(V_1)$, $b'' \in \bc_*(X\setmin V_1)$, -$e' \in \bc_*(p(V_1))$, and $e'' \in \bc_*(p(X\setmin V_1))$. +where $p' \in CD_*(V)$, $p'' \in CD_*(X\setmin V)$, +$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, +$e' \in \bc_*(p(V))$, and $e'' \in \bc_*(p(X\setmin V))$. (Note that since the family of diffeomorphisms $p$ is constant (independent of parameters) -near $\bd V_1)$, the expressions $p(V_1) \sub X$ and $p(X\setmin V_1) \sub X$ are +near $\bd V)$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are unambiguous.) We also have that $\deg(b'') = 0 = \deg(p'')$. -Choose $x' \in \bc_*(p(V_1))$ such that $\bd x' = f'$. +Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$. This is possible by \nn{...}. Finally, define \[ e(p\ot b) \deq x' \bullet p''(b'') . \] +Note that above we are essentially using the method of acyclic models. +For each generator $p\ot b$ we specify the acyclic (in positive degrees) +target complex $\bc_*(p(V)) \bullet p''(b'')$. + +The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices: +The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$. +The next two lemmas show that up to (iterated) homotopy $e$ is independent +of these choices. + +\begin{lemma} +Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with +different choices of $x'$ at each step. +(Same choice of $V$ at each step.) +Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$. +Any two choices of such a first-order homotopy are second-order homotopic, and so on, +to arbitrary order. +\end{lemma} + +\begin{proof} +This is a standard result in the method of acyclic models. +\nn{should we say more here?} +\nn{maybe this lemma should be subsumed into the next lemma. probably it should.} +\end{proof} + +\begin{lemma} +Let $\tilde{e} : G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with +different choices of $V$ (and hence also different choices of $x'$) at each step. +If $m \ge 1$ then $e$ and $\tilde{e}$ are homotopic. +If $m \ge 2$ then any two choices of this first-order homotopy are second-order homotopic. +And so on. +In other words, $e : G_*^{i,m} \to \bc_*(X)$ is well-defined up to $m$-th order homotopy. +\end{lemma} + +\begin{proof} +We construct $h: G_*^{i,m} \to \bc_*(X)$ such that $\bd h + h\bd = e - \tilde{e}$. +$e$ and $\tilde{e}$ coincide on bidegrees $(0, j)$, so define $h$ +to be zero there. +Assume inductively that $h$ has been defined for degrees less than $k$. +Let $p\ot b$ be a generator of degree $k$. +Choose $V_1$ as in the definition of $G_*^{i,m}$ so that +\[ + N_{i,k+1}(p\ot b) \subeq V_1 \subeq N_{i,k+2}(p\ot b) . +\] +There are splittings +\[ + p = p'_1\bullet p''_1 , \;\; b = b'_1\bullet b''_1 , + \;\; e(p\ot b) - \tilde{e}(p\ot b) - h(\bd(p\ot b)) = f'_1\bullet f''_1 , +\] +where $p'_1 \in CD_*(V_1)$, $p''_1 \in CD_*(X\setmin V_1)$, +$b'_1 \in \bc_*(V_1)$, $b''_1 \in \bc_*(X\setmin V_1)$, +$f'_1 \in \bc_*(p(V_1))$, and $f''_1 \in \bc_*(p(X\setmin V_1))$. +Inductively, $\bd f'_1 = 0$. +Choose $x'_1 \in \bc_*(p(V_1))$ so that $\bd x'_1 = f'_1$. +Define +\[ + h(p\ot b) \deq x'_1 \bullet p''_1(b''_1) . +\] +This completes the construction of the first-order homotopy when $m \ge 1$. + +The $j$-th order homotopy is constructed similarly, with $V_j$ replacing $V_1$ above. +\end{proof} + +Note that on $G_*^{i,m+1} \subeq G_*^{i,m}$, we have defined two maps, +call them $e_{i,m}$ and $e_{i,m+1}$. +An easy variation on the above lemma shows that $e_{i,m}$ and $e_{i,m+1}$ are $m$-th +order homotopic. + \medskip +\noop{ + + +\begin{lemma} + +\end{lemma} +\begin{proof} + +\end{proof} + + +} + + \nn{to be continued....}