# HG changeset patch # User kevin@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1214502980 0 # Node ID aac9fd8d6bc62e133fcdd9ca5958cdddd2405996 # Parent c73e8beb4a20987322b3145b402a94bce2a29cc9 finished with evaluation map stuff for now. diff -r c73e8beb4a20 -r aac9fd8d6bc6 blob1.tex --- a/blob1.tex Tue Jun 24 19:46:06 2008 +0000 +++ b/blob1.tex Thu Jun 26 17:56:20 2008 +0000 @@ -607,6 +607,8 @@ Any other map satisfying the above two properties is homotopic to $e_X$. \end{prop} +\nn{need to rewrite for self-gluing instead of gluing two pieces together} + \nn{Should say something stronger about uniqueness. Something like: there is a contractible subcomplex of the complex of chain maps @@ -622,6 +624,7 @@ \medskip The proof will occupy the remainder of this section. +\nn{unless we put associativity prop at end} \medskip @@ -656,6 +659,7 @@ \begin{lemma} \label{extension_lemma} Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$. Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. +Furthermore, one can choose the homotopy so that its support is equal to the support of $x$. \end{lemma} The proof will be given in Section \ref{fam_diff_sect}. @@ -722,55 +726,59 @@ a homotopy (rel $\bd x$) to $x' \in G_*$, and further that $x'$ and this homotopy are unique up to iterated homotopy. - - - +Given $k>0$ and a blob diagram $b$, we say that a cover of $X$ $\cU$ is $k$-compatible with +$b$ if, for any $\{U_1, \ldots, U_k\} \sub \cU$, the union +$U_1\cup\cdots\cup U_k$ is a union of balls which satisfies the condition used to define $G_*$ above. +Note that if a family of diffeomorphisms $p$ is adapted to +$\cU$ and $b$ is a blob diagram occurring in $x$, then $p\otimes b \in G_*$. +\nn{maybe emphasize this more; it's one of the main ideas in the proof} -\medskip -\hrule -\medskip -\hrule -\medskip -\nn{older stuff:} +Let $k$ be the degree of $x$ and choose a cover $\cU$ of $X$ such that $\cU$ is +$k$-compatible with each of the (finitely many) blob diagrams occurring in $x$. +We will use Lemma \ref{extension_lemma} with respect to the cover $\cU$ to +construct the homotopy to $G_*$. +First we will construct a homotopy $h \in G_*$ from $\bd x$ to a cycle $z$ such that +each family of diffeomorphisms $p$ occurring in $z$ is adapted to $\cU$. +Then we will construct a homotopy (rel boundary) $r$ from $x + h$ to $y$ such that +each family of diffeomorphisms $p$ occurring in $y$ is adapted to $\cU$. +This implies that $y \in G_*$. +$r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. -Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ -(e.g.~the support of a blob diagram). -We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if -$f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ -either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. -A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, -each of which is compatible. -(Note that we could strengthen the definition of compatibility to incorporate -a factorization condition, similar to the definition of ``adapted to" above. -The weaker definition given here will suffice for our needs below.) +We will define $h$ inductive on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. +Define $h$ to be zero on bidegree $(0, k-1)$. +Let $p\otimes b$ be a generator occurring in $\bd x$ with bidegree $(1, k-2)$. +Using Lemma \ref{extension_lemma}, construct a homotopy $q$ from $p$ to $p'$ which is adapted to $\cU$. +Define $h$ at $p\otimes b$ to be $q\otimes b$. +Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$. +Let $a$ be that portion of $\bd(p'\otimes b')$ which intersects the boundary of +bidegree $(1, k-2)$ stuff. +Apply Lemma \ref{extension_lemma} to $p'$ plus the diffeo part of $h(a)$ +(rel the outer boundary of said part), +yielding a family of diffeos $q'$. +\nn{definitely need to say this better} +Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$. +Continuing in this way, we define all of $h$. -\begin{cor} \label{extension_lemma_2} -Let $x \in CD_k(X)$ be a singular chain such that $\bd x$ is compatible with $\{B_j\}$. -Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. -\end{cor} -\begin{proof} -This will follow from Lemma \ref{extension_lemma} for -appropriate choice of cover $\cU = \{U_\alpha\}$. -Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let -$V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. -Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ -either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. +The homotopy $r$ is constructed similarly. + +\nn{need to say something about uniqueness of $r$, $h$ etc. +postpone this until second draft.} -Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, -with the (compatible) support of $f_i$ in place of $X$. -This insures that the resulting homotopy $h_i$ is compatible. -Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. -\nn{actually, need to start with the 0-skeleton of $\bd x$, then 1-skeleton, etc.; fix this} -\end{proof} +At this point, we have finished defining the evaluation map. +The uniqueness statement in the proposition is clear from the method of proof. +All that remains is to show that the evaluation map gets along well with cutting and gluing, +as claimed in the proposition. +This is in fact not difficult, since the myriad choices involved in defining the +evaluation map can be made in parallel for the top and bottom +arrows in the commutative diagram. + +This completes the proof of Proposition \ref{CDprop}. \medskip -((argument continues roughly as follows: up to homotopy, there is only one way to define $e_X$ -on compatible $x\otimes y \in CD_*(X)\otimes \bc_*(X)$. -This is because $x$ is the gluing of $x'$ and $x''$, where $x'$ has degree zero and is defined on -the complement of the $D_i$'s, and $x''$ is defined on the $D_i$'s. -We have no choice on $x'$, since we already know the map on 0-parameter families of diffeomorphisms. -We have no choice, up to homotopy, on $x''$, since the target chain complex is contractible.)) +\nn{say something about associativity here} + + \section{Families of Diffeomorphisms} \label{fam_diff_sect}