diff -r 123a8b83e02c -r 9caa4d68a8a5 text/evmap.tex --- a/text/evmap.tex Sun Sep 19 22:29:29 2010 -0500 +++ b/text/evmap.tex Sun Sep 19 22:57:10 2010 -0500 @@ -21,7 +21,7 @@ introduce a homotopy equivalent alternate version of the blob complex, $\btc_*(X)$, which is more amenable to this sort of action. Recall from Remark \ref{blobsset-remark} that blob diagrams -have the structure of a sort-of-simplicial set. +have the structure of a sort-of-simplicial set. \nn{need a more conventional sounding name: `polyhedral set'?} Blob diagrams can also be equipped with a natural topology, which converts this sort-of-simplicial set into a sort-of-simplicial space. Taking singular chains of this space we get $\btc_*(X)$. @@ -70,17 +70,17 @@ \medskip Fix $\cU$, an open cover of $X$. -Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ +Define the ``small blob complex" $\bc^{\cU}_*(X)$ to be the subcomplex of $\bc_*(X)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$. \begin{lemma}[Small blobs] \label{small-blobs-b} \label{thm:small-blobs} -The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. +The inclusion $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence. \end{lemma} \begin{proof} -It suffices to show that for any finitely generated pair of subcomplexes +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 \[ (C_*, D_*) \sub (\bc_*(X), \sbc_*(X)) \] @@ -92,19 +92,19 @@ for all $x\in C_*$. For simplicity we will assume that all fields are splittable into small pieces, so that -$\sbc_0(X) = \bc_0$. +$\sbc_0(X) = \bc_0(X)$. (This is true for all of the examples presented in this paper.) Accordingly, we define $h_0 = 0$. Next we define $h_1$. Let $b\in C_1$ be a 1-blob diagram. Let $B$ be the blob of $b$. -We will construct a 1-chain $s(b)\in \sbc_1$ such that $\bd(s(b)) = \bd b$ +We will construct a 1-chain $s(b)\in \sbc_1(X)$ such that $\bd(s(b)) = \bd b$ and the support of $s(b)$ is contained in $B$. -(If $B$ is not embedded in $X$, then we implicitly work in some term of a decomposition +(If $B$ is not embedded in $X$, then we implicitly work in some stage of a decomposition of $X$ where $B$ is embedded. -See \ref{defn:configuration} and preceding discussion.) -It then follows from \ref{disj-union-contract} that we can choose +See Definition \ref{defn:configuration} and preceding discussion.) +It then follows from Corollary \ref{disj-union-contract} that we can choose $h_1(b) \in \bc_1(X)$ such that $\bd(h_1(b)) = s(b) - b$. Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series @@ -113,9 +113,9 @@ Let $\cV_1$ be an auxiliary open cover of $X$, subordinate to $\cU$ and also satisfying conditions specified below. -Let $b = (B, u, r)$, $u = \sum a_i$ be the label of $B$, $a_i\in \bc_0(B)$. +Let $b = (B, u, r)$, with $u = \sum a_i$ the label of $B$, and $a_i\in \bc_0(B)$. Choose a sequence of collar maps $\bar{f}_j:B\cup\text{collar}\to B$ satisfying conditions which we cannot express -until introducing more notation. +until introducing more notation. \nn{needs some rewriting, I guess} Let $f_j:B\to B$ be the restriction of $\bar{f}_j$ to $B$; $f_j$ maps $B$ homeomorphically to a slightly smaller submanifold of $B$. Let $g_j = f_1\circ f_2\circ\cdots\circ f_j$. @@ -125,7 +125,7 @@ $g_{j-1}(|f_j|)$ is also contained is an open set of $\cV_1$. There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$ -(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) +(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$ \nn{doesn't strictly make any sense}) and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. Define \[ @@ -141,12 +141,12 @@ Let $B = |b|$, either a ball or a union of two balls. By possibly working in a decomposition of $X$, we may assume that the ball(s) of $B$ are disjointly embedded. -We will construct a 2-chain $s(b)\in \sbc_2$ such that +We will construct a 2-chain $s(b)\in \sbc_2(X)$ such that \[ \bd(s(b)) = \bd(h_1(\bd b) + b) = s(\bd b) \] and the support of $s(b)$ is contained in $B$. -It then follows from \ref{disj-union-contract} that we can choose +It then follows from Corollary \ref{disj-union-contract} that we can choose $h_2(b) \in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$. Similarly to the construction of $h_1$ above, @@ -156,7 +156,7 @@ disjoint union of balls. Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and -also satisfying conditions specified below. +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 @@ -168,7 +168,7 @@ Let $s(\bd b) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams appearing in the boundaries of the $e_k$. As in the construction of $h_1$, we can choose 1-blob diagrams $q_m$ such that -$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. +$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ and $|q_m|$ is contained in an open set of $\cV_1$. If $x$ is a sum of $p_m$'s, we denote the corresponding sum of $q_m$'s by $q(x)$. Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. @@ -183,7 +183,7 @@ (In this case there are either one or two balls in the disjoint union.) For any fixed open cover $\cV_2$ this condition can be satisfied by choosing $\cV_1$ to be a sufficiently fine cover. -It follows from \ref{disj-union-contract} that we can choose +It follows from Corollary \ref{disj-union-contract} that we can choose $x_k \in \bc_2(X)$ with $\bd x_k = g_{j-1}(e_k) - g_j(e_k) - q(\bd e_k)$ and with $\supp(x_k) = U$. We can now take $d_j \deq \sum x_k$. @@ -219,24 +219,25 @@ We give $\BD_k$ the finest topology such that \begin{itemize} \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 $\BD_0(B)$. +$\bc_0(B)$ 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 -$P\to \BD_k(M)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map -$P\to \Homeo(M)$, with the twig blob labels $u_i(p)$ varying independently. +$P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map +$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, if we did allow this it would not affect the truth of the claims we make below. -In particular, we would get a homotopy equivalent complex $\btc_*(M)$. +In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) whose $(i,j)$ entry is $C_j(\BD_i)$, the singular $j$-chains on the space of $i$-blob diagrams. The vertical boundary of the double complex, denoted $\bd_t$, is the singular boundary, and the horizontal boundary, denoted $\bd_b$, is -the blob boundary. +the blob boundary. Following the usual sign convention, we have $\bd = \bd_b + (-1)^i \bd_t$. We will regard $\bc_*(X)$ as the subcomplex $\btc_{*0}(X) \sub \btc_{**}(X)$. The main result of this subsection is @@ -266,9 +267,10 @@ e: \btc_{ij}\to\btc_{i+1,j} \] 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 $r\circ y : P\to \BD_0$. +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 @@ -304,12 +306,14 @@ &= 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?}. + For $x\in \btc_{00}$ we have \nn{ignoring signs} \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)\\ - &= x - r(x). + &= x - r(x). \qedhere \end{align*} \end{proof} @@ -317,10 +321,10 @@ For manifolds $X$ and $Y$, we have $\btc_*(X\du Y) \simeq \btc_*(X)\otimes\btc_*(Y)$. \end{lemma} \begin{proof} -This follows from the Eilenber-Zilber theorem and the fact that -\[ - \BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . -\] +This follows from the Eilenberg-Zilber theorem and the fact that +\begin{align*} + \BD_k(X\du Y) & \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) . \qedhere +\end{align*} \end{proof} For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$} @@ -358,13 +362,13 @@ \end{proof} -\begin{proof}[Proof of \ref{lem:bc-btc}] -Armed with the above lemmas, we can now proceed similarly to the proof of \ref{small-blobs-b}. +\begin{proof}[Proof of Lemma \ref{lem:bc-btc}] +Armed with the above lemmas, we can now proceed similarly to the proof of Lemma \ref{small-blobs-b}. It suffices to show that for any finitely generated pair of subcomplexes $(C_*, D_*) \sub (\btc_*(X), \bc_*(X))$ we can find a homotopy $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$ -and $x + h\bd(x) + \bd h(X) \in \bc_*(X)$ for all $x\in C_*$. +and $x + h\bd(x) + \bd h(x) \in \bc_*(X)$ for all $x\in C_*$. By Lemma \ref{small-top-blobs}, we may assume that $C_* \sub \btc_*^\cU(X)$ for some cover $\cU$ of our choosing. @@ -376,22 +380,22 @@ Let $b \in C_1$ be a generator. Since $b$ is supported in a disjoint union of balls, we can find $s(b)\in \bc_1$ with $\bd (s(b)) = \bd b$ -(by \ref{disj-union-contract}), and also $h_1(b) \in \btc_2$ +(by Corollary \ref{disj-union-contract}), and also $h_1(b) \in \btc_2(X)$ such that $\bd (h_1(b)) = s(b) - b$ -(by \ref{bt-contract} and \ref{btc-prod}). +(by Lemmas \ref{bt-contract} and \ref{btc-prod}). Now let $b$ be a generator of $C_2$. If $\cU$ is fine enough, there is a disjoint union of balls $V$ on which $b + h_1(\bd b)$ is supported. -Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find -$s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}). -By \ref{bt-contract} and \ref{btc-prod}, we can now find -$h_2(b) \in \btc_3$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ +Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2(X)$, we can find +$s(b)\in \bc_2(X)$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by Corollary \ref{disj-union-contract}). +By Lemmas \ref{bt-contract} and \ref{btc-prod}, we can now find +$h_2(b) \in \btc_3(X)$, also supported on $V$, such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$ The general case, $h_k$, is similar. \end{proof} -The proof of \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion +The proof of Lemma \ref{lem:bc-btc} constructs a homotopy inverse to the inclusion $\bc_*(X)\sub \btc_*(X)$. One might ask for more: a contractible set of possible homotopy inverses, or at least an $m$-connected set for arbitrarily large $m$. @@ -440,7 +444,7 @@ \begin{proof} In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with $\bc_*$ replaced by $\btc_*$. -And in fact for $\btc_*$ we get a sharper result: we can omit +In fact, for $\btc_*$ we get a sharper result: we can omit the ``up to homotopy" qualifiers. Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$,