diff -r aac9fd8d6bc6 -r ea489bbccfbf blob1.tex --- a/blob1.tex Thu Jun 26 17:56:20 2008 +0000 +++ b/blob1.tex Fri Jun 27 04:24:25 2008 +0000 @@ -611,11 +611,11 @@ \nn{Should say something stronger about uniqueness. Something like: there is -a contractible subcomplex of the complex of chain maps +a contractible subcomplex of the complex of chain maps $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (0-cells are the maps, 1-cells are homotopies, etc.), and all choices in the construction lie in the 0-cells of this contractible subcomplex. -Or maybe better to say any two choices are homotopic, and +Or maybe better to say any two choices are homotopic, and any two homotopies and second order homotopic, and so on.} \nn{Also need to say something about associativity. @@ -669,7 +669,7 @@ The strategy for the proof of Proposition \ref{CDprop} is as follows. We will identify a subcomplex \[ - G_* \sub CD_*(X) \otimes \bc_*(X) + G_* \sub CD_*(X) \otimes \bc_*(X) \] on which the evaluation map is uniquely determined (up to homotopy) by the conditions in \ref{CDprop}. @@ -678,12 +678,12 @@ \nn{need to be more specific here} Let $p$ be a singular cell in $CD_*(X)$ and $b$ be a blob diagram in $\bc_*(X)$. -Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ +Roughly speaking, $p\otimes b$ is in $G_*$ if each component $V$ of the support of $p$ intersects at most one blob $B$ of $b$. Since $V \cup B$ might not itself be a ball, we need a more careful and complicated definition. Choose a metric for $X$. -We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that -$N_\epsilon(b) \cup \supp(p)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon +We define $p\otimes b$ to be in $G_*$ if there exist $\epsilon > 0$ such that +$\supp(p) \cup N_\epsilon(b)$ is a union of balls, where $N_\epsilon(b)$ denotes the epsilon neighborhood of the support of $b$. \nn{maybe also require that $N_\delta(b)$ is a union of balls for all $\delta<\epsilon$.} @@ -704,15 +704,16 @@ let $p\otimes b$ be a generator of $G_k$. Let $C \sub X$ be a union of balls (as described above) containing $\supp(p)\cup\supp(b)$. There is a factorization $p = p' \circ g$, where $g\in \Diff(X)$ and $p'$ is a family of diffeomorphisms which is the identity outside of $C$. +\scott{Shouldn't this be $p = g\circ p'$?} Let $b = b'\bullet b''$, where $b' \in \bc_*(C)$ and $b'' \in \bc_0(X\setmin C)$. -We may assume inductively that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where +We may assume inductively \scott{why? I don't get this.} that $e_X(\bd(p\otimes b))$ has the form $x\bullet g(b'')$, where $x \in \bc_*(g(C))$. Since $\bc_*(g(C))$ is contractible, there exists $y \in \bc_*(g(C))$ such that $\bd y = x$. \nn{need to say more if degree of $x$ is 0} Define $e_X(p\otimes b) = y\bullet g(b'')$. We now show that $e_X$ on $G_*$ is, up to homotopy, independent of the various choices made. -If we make a different series of choice of the chain $y$ in the previous paragraph, +If we make a different series of choice of the chain $y$ in the previous paragraph, we can inductively construct a homotopy between the two sets of choices, again relying on the contractibility of $\bc_*(g(G))$. A similar argument shows that this homotopy is unique up to second order homotopy, and so on. @@ -726,25 +727,25 @@ 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 +Given $k>0$ and a blob diagram $b$, we say that a cover $\cU$ of $X$ 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 $p\otimes b \in G_*$. +Note that if a family of diffeomorphisms $p$ is adapted to +$\cU$ and $b$ is a blob diagram occurring in $x$ \scott{huh, what's $x$ here?}, then $p\otimes b \in G_*$. \nn{maybe emphasize this more; it's one of the main ideas in the proof} 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 +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. +The homotopy $r$ can also be thought of as a homotopy from $x$ to $y-h \in G_*$, and this is the homotopy we seek. -We will define $h$ inductive on bidegrees $(0, k-1), (1, k-2), \ldots, (k-1, 0)$. +We will define $h$ inductively 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 $\cU$. @@ -761,7 +762,7 @@ The homotopy $r$ is constructed similarly. -\nn{need to say something about uniqueness of $r$, $h$ etc. +\nn{need to say something about uniqueness of $r$, $h$ etc. postpone this until second draft.} At this point, we have finished defining the evaluation map.