blob: comparison text/evmap.tex

equal deleted inserted replaced

-:0bc6fa29b62a
+:24be062a87a1
 \begin{lemma}[Small blobs] \label{small-blobs-b}  \label{thm:small-blobs}
 The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence.
 \end{lemma}
 \begin{proof}
-It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that
+It suffices \nn{why? we should spell this out somewhere} to show that for any finitely generated
+pair $(C_*, D_*)$, with $D_*$ a subcomplex of $C_*$ such that
 \[
 	(C_*, D_*) \sub (\bc_*(X), \sbc_*(X))
 \]
 we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$
 and
 of small collar maps, plus a shrunken version of $b$.
 The composition of all the collar maps shrinks $B$ to a sufficiently small
 disjoint union of balls.
 Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and
-also satisfying conditions specified below. \nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.}
+also satisfying conditions specified below.
+\nn{This happens sufficiently far below (i.e. not in this paragraph) that we probably should give better warning.}
 As before, choose a sequence of collar maps $f_j$
 such that each has support
 contained in an open set of $\cV_1$ and the composition of the corresponding collar homeomorphisms
 yields an embedding $g:B\to B$ such that $g(B)$ is contained in an open set of $\cV_1$.
 Let $g_j:B\to B$ be the embedding at the $j$-th stage.
 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous.
 \item \nn{don't we need something for collaring maps?}
 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous.
 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous,
 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on
-$\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
+$\bc_0(B)$ comes from the generating set $\BD_0(B)$.
+\nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space}
 \end{itemize}
 We can summarize the above by saying that in the typical continuous family
 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map
 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently.
 \]
 adds an outermost blob, equal to all of $B^n$, to the $j$-parameter family of blob diagrams.
 Note that for fixed $i$, $e$ is a chain map, i.e. $\bd_t e = e \bd_t$.
 A generator $y\in \btc_{0j}$ is a map $y:P\to \BD_0$, where $P$ is some $j$-dimensional polyhedron.
-We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$. \nn{I found it pretty confusing to reuse the letter $r$ here.}
+We define $r(y)\in \btc_{0j}$ to be the constant function $r\circ y : P\to \BD_0$.
+\nn{I found it pretty confusing to reuse the letter $r$ here.}
 Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking
 the same value (namely $r(y(p))$, for any $p\in P$).
 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams
 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$.
 Now define, for $y\in \btc_{0j}$,
 											e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\
 			&= x - r(x) - \bd_t(c(r(x))) + c(r(\bd_t x)) \\
 			&= x - r(x) + r(x) \\
 			&= x.
 \end{align*}
-Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ \nn{explain why this is true?} and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
+Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram,
+as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$
+\nn{explain why this is true?}
+and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}.
 For $x\in \btc_{00}$ we have
 \begin{align*}
 	\bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\
 			&= x - r(x) + r(x) - r(x)\\
 CH_*(X\sgl, Y\sgl) \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}}   & 	\bc_*(Y\sgl)
 }
 \end{equation*}
 \end{enumerate}
 Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$
-satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy.
+satisfying the above two conditions which is $m$-connected. In particular,
+this means that the choice of chain map above is unique up to homotopy.
 \end{thm}
 \begin{rem}
-Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this.
+Note that the statement doesn't quite give uniqueness up to iterated homotopy.
+We fully expect that this should actually be the case, but haven't been able to prove this.
 \end{rem}
 Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma},
 and then give an outline of the method of proof.
 near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are
 unambiguous.)
 We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.
 %We also have that $\deg(b'') = 0 = \deg(p'')$.
 Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.
-This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility}  and the fact that isotopic fields
+This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility}
-differ by a local relation.
+and the fact that isotopic fields differ by a local relation.
 Finally, define
 \[
 	e(p\ot b) \deq x' \bullet p''(b'') .
 \]
 for all $l\ge m$ and all $q\ot a$ which appear in the boundary of $p\ot b$.
 \begin{proof}
-There exists $\lambda > 0$ such that for every  subset $c$ of the blobs of $b$ the set $\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
+There exists $\lambda > 0$ such that for every  subset $c$ of the blobs of $b$ the set
+$\Nbd_u(c)$ is homeomorphic to $|c|$ for all $u < \lambda$ .
 (Here we are using the fact that the blobs are
 piecewise smooth or piecewise-linear and that $\bd c$ is collared.)
 We need to consider all such $c$ because all generators appearing in
 iterated boundaries of $p\ot b$ must be in $G_*^{i,m}$.)

changeset 544	24be062a87a1
parent 541	4f142fcd386e
child 545	8f33a46597c4