blob: comparison blob1.tex

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 \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
 $CD_*(X) \otimes \bc_*(X) \to \bc_*(X)$ (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
 any two homotopies and second order homotopic, and so on.}
 \nn{Also need to say something about associativity.
 Put it in the above prop or make it a separate prop?
 I lean toward the latter.}
 \medskip
 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}.
 We then show that the inclusion of $G_*$ into the full complex
 is an equivalence in the appropriate sense.
 \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 $p$
 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
+$\supp(p) \cup N_\epsilon(b)$ 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 $\delta<\epsilon$.}
 \nn{need to worry about case where the intrinsic support of $p$ is not a union of balls}
 because $G_0 \sub CD_0\otimes \bc_0$.
 Assume we have defined the evaluation map up to $G_{k-1}$ and
 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,
 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.
 Given a different set of choices $\{C'\}$ of the unions of balls $\{C\}$,
 Next we show that given $x \in CD_*(X) \otimes \bc_*(X)$ with $\bd x \in G_*$, there exists
 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
+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 $p\otimes b \in G_*$.
+$\cU$ 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
 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 $\cU$.
 Define $h$ at $p\otimes b$ to be $q\otimes b$.
 Let $p'\otimes b'$ be a generator occurring in $\bd x$ with bidegree $(2, k-3)$.
 Define $h$ at $p'\otimes b'$ to be $q'\otimes b'$.
 Continuing in this way, we define all of $h$.
 The homotopy $r$ is constructed similarly.
 \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.
 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,

changeset 19	ea489bbccfbf
parent 18	aac9fd8d6bc6
child 20	60a068a5be10