# HG changeset patch # User Kevin Walker # Date 1283183641 25200 # Node ID 803cc581fd42d00f6874e1d0e13e44bbee124eb8 # Parent 4e4b6505d9ef4990d4e5101e852f1dfc6d202778 revisions of evmap; misc diff -r 4e4b6505d9ef -r 803cc581fd42 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sat Aug 28 17:34:20 2010 -0700 +++ b/text/a_inf_blob.tex Mon Aug 30 08:54:01 2010 -0700 @@ -288,26 +288,27 @@ or $M\to Y$, instead of an undecorated colimit with fancier $k$-categories over $Y$. Information about the specific map to $Y$ has been taken out of the categories and put into sphere modules and decorations. - -Let $F \to E \to Y$ be a fiber bundle as above. -Choose a decomposition $Y = \cup X_i$ -such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. -\nn{resume revising here} -Choose the product structure (trivialization of the bundle restricted to $X_i$) as well. -To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). -And more generally to each codim-$j$ face we have an $S^{j-1}$-module. -Decorate the decomposition with these modules and do the colimit. +\nn{...} - -\nn{There is a version of this last construction for arbitrary maps $E \to Y$ -(not necessarily a fibration). -In fact, there is also a version of the first construction for non-fibrations.} - - - -Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. -Let $X_1$ and $X_2$ be $n$-manifolds - +%Let $F \to E \to Y$ be a fiber bundle as above. +%Choose a decomposition $Y = \cup X_i$ +%such that the restriction of $E$ to $X_i$ is a product $F\times X_i$, +%and choose trivializations of these products as well. +% +%\nn{edit marker} +%To each codim-1 face $X_i\cap X_j$ we have a bimodule ($S^0$-module). +%And more generally to each codim-$j$ face we have an $S^{j-1}$-module. +%Decorate the decomposition with these modules and do the colimit. +% +% +%\nn{There is a version of this last construction for arbitrary maps $E \to Y$ +%(not necessarily a fibration).} +% +% +% +%Note that Theorem \ref{thm:gluing} can be viewed as a special case of this one. +%Let $X_1$ and $X_2$ be $n$-manifolds +% \subsection{A gluing theorem} diff -r 4e4b6505d9ef -r 803cc581fd42 text/evmap.tex --- a/text/evmap.tex Sat Aug 28 17:34:20 2010 -0700 +++ b/text/evmap.tex Mon Aug 30 08:54:01 2010 -0700 @@ -13,11 +13,12 @@ where $CH_*(X, Y) = C_*(\Homeo(X, Y))$, the singular chains on the space of homeomorphisms from $X$ to $Y$. (If $X$ and $Y$ have non-empty boundary, these families of homeomorphisms -are required to be fixed on the boundaries.) +are required to restrict to a fixed homeomorphism on the boundaries.) +These actions (for various $X$ and $Y$) are compatible with gluing. See \S \ref{ss:emap-def} for a more precise statement. The most convenient way to prove that maps $e_{XY}$ with the desired properties exist is to -introduce a homotopy equivalent alternate version of the blob complex $\btc_*(X)$ +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. @@ -25,8 +26,10 @@ sort-of-simplicial set into a sort-of-simplicial space. Taking singular chains of this space we get $\btc_*(X)$. The details are in \S \ref{ss:alt-def}. -For technical reasons we also show that requiring the blobs to be -embedded yields a homotopy equivalent complex. +We also prove a useful lemma (\ref{small-blobs-b}) which says that we can assume that +blobs are small with respect to any fixed open cover. + + %Since $\bc_*(X)$ and $\btc_*(X)$ are homotopy equivalent one could try to construct %the $CH_*$ actions directly in terms of $\bc_*(X)$. @@ -48,11 +51,14 @@ If $b$ is a blob diagram in $\bc_*(X)$, define the {\it support} of $b$, denoted $\supp(b)$ or $|b|$, to be the union of the blobs of $b$. -For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union -of the supports of the blob diagrams which appear in it. +%For a general $k$-chain $a\in \bc_k(X)$, define the support of $a$ to be the union +%of the supports of the blob diagrams which appear in it. +More generally, we say that a chain $a\in \bc_k(X)$ is supported on $S$ if +$a = a'\bullet r$, where $a'\in \bc_k(S)$ and $r\in \bc_0(X\setmin S)$. -If $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is +Similarly, if $f: P\times X\to X$ is a family of homeomorphisms and $Y\sub X$, we say that $f$ is {\it supported on $Y$} if $f(p, x) = f(p', x)$ for all $x\in X\setmin Y$ and all $p,p'\in P$. +%Equivalently, $f = f'\bullet r$, where $f'\in CH_k(Y)$ and $r\in CH_0(X\setmin Y)$. We will sometimes abuse language and talk about ``the" support of $f$, again denoted $\supp(f)$ or $|f|$, to mean some particular choice of $Y$ such that $f$ is supported on $Y$. @@ -61,6 +67,8 @@ (cf. end of \S \ref{ss:syst-o-fields}), we say that $f$ is supported on $S\sub M$ if $f(x) = x$ for all $x\in M\setmin S$. +\medskip + Fix $\cU$, an open cover of $X$. Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(X)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, @@ -79,7 +87,7 @@ we can find a homotopy $h:C_*\to \bc_*(X)$ such that $h(D_*) \sub \sbc_*(X)$ and \[ - x + h\bd(x) + \bd h(X) \in \sbc_*(X) + h\bd(x) + \bd h(x) - x \in \sbc_*(X) \] for all $x\in C_*$. @@ -101,18 +109,24 @@ Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series of small collar maps, plus a shrunken version of $b$. -The composition of all the collar maps shrinks $B$ to a sufficiently small ball. +The composition of all the collar maps shrinks $B$ to a ball which is small with respect to $\cU$. 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 $B$, $a_i\in \bc_0(B)$. -Choose a sequence of collar maps $f_j:\bc_0(B)\to\bc_0(B)$ 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 $\cV_1$. -\nn{need to say this better; maybe give fig} -Let $g_j:B\to B$ be the embedding at the $j$-th stage. +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. +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 $g_j = f_1\circ f_2\circ\cdots\circ f_j$. +Let $g$ be the last of the $g_j$'s. +Choose the sequence $\bar{f}_j$ so that +$g(B)$ is contained is an open set of $\cV_1$ and +$g_{j-1}(|f_j|)$ 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 $\cV_1$ -and $\bd c_{ij} = g_j(a_i) = g_{j-1}(a_i)$. +(more specifically, $|c_{ij}| = g_{j-1}(|f_j|)$) +and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$. Define \[ s(b) = \sum_{i,j} c_{ij} + g(b) @@ -150,23 +164,18 @@ Let $g_j:B\to B$ be the embedding at the $j$-th stage. Fix $j$. -We will construct a 2-chain $d_j$ such that $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$. -Let $g_{j-1}(s(\bd b)) = \sum e_k$, and let $\{p_m\}$ be the 0-blob diagrams +We will construct a 2-chain $d_j$ such that $\bd d_j = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$. +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 = f_j(p_m) - p_m$ and $\supp(q_m)$ is contained in an open set of $\cV_1$. -%%% \nn{better not to do this, to make things more parallel with general case (?)} -%Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support -%is contained in a open set of $\cV_1$. -%(This is possible since there are only finitely many $p_m$.) +$\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$. 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$, $e_k + q(\bd e_k)$. -This is a 1-chain whose boundary is $f_j(\bd e_k)$. +Now consider, for each $k$, $g_{j-1}(e_k) - q(\bd e_k)$. +This is a 1-chain whose boundary is $g_j(\bd e_k)$. The support of $e_k$ is $g_{j-1}(V)$ for some $V\in \cV_1$, and the support of $q(\bd e_k)$ is contained in a union $V'$ of finitely many open sets of $\cV_1$, all of which contain the support of $f_j$. -%the support of $q(\bd e_k)$ is contained in $V'$ for some $V'\in \cV_1$. We now reveal the mysterious condition (mentioned above) which $\cV_1$ satisfies: the union of $g_{j-1}(V)$ and $V'$, for all of the finitely many instances arising in the construction of $h_2$, lies inside a disjoint union of balls $U$ @@ -174,11 +183,11 @@ (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 $x_k \in \bc_2(X)$ with $\bd x_k = f_j(e_k) - e_k - q(\bd e_k)$ +It follows from \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$. -It is clear that $\bd d_j = \sum (f_j(e_k) - e_k) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$, as desired. +It is clear that $\bd d_j = \sum (g_{j-1}(e_k) - g_j(e_k)) = g_{j-1}(s(\bd b)) - g_{j}(s(\bd b))$, as desired. \nn{should maybe have figure} We now define diff -r 4e4b6505d9ef -r 803cc581fd42 text/intro.tex --- a/text/intro.tex Sat Aug 28 17:34:20 2010 -0700 +++ b/text/intro.tex Mon Aug 30 08:54:01 2010 -0700 @@ -411,7 +411,7 @@ \] \end{thm:product} The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps -(see \S \ref{moddecss}). +(see \S \ref{ss:product-formula}). Fix a topological $n$-category $\cC$, which we'll omit from the notation. Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.