diff -r 352389c6ddcf -r edf8798ef477 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Fri Aug 27 10:58:21 2010 -0700 +++ b/text/a_inf_blob.tex Fri Aug 27 15:36:21 2010 -0700 @@ -12,19 +12,19 @@ $\cl{\cC}(M)$ is homotopy equivalent to our original definition of the blob complex $\bc_*^\cD(M)$. -\medskip +%\medskip -An important technical tool in the proofs of this section is provided by the idea of ``small blobs". -Fix $\cU$, an open cover of $M$. -Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ -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{thm}[Small blobs] \label{thm:small-blobs} -The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. -\end{thm} -The proof appears in \S \ref{appendix:small-blobs}. +%An important technical tool in the proofs of this section is provided by the idea of ``small blobs". +%Fix $\cU$, an open cover of $M$. +%Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ +%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{thm}[Small blobs] \label{thm:small-blobs} +%The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. +%\end{thm} +%The proof appears in \S \ref{appendix:small-blobs}. \subsection{A product formula} \label{ss:product-formula} @@ -69,7 +69,7 @@ Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. -It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ +It follows from Lemma \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. (If the blobs of $a$ are small with respect to a sufficiently fine cover then their projections to $Y$ are contained in some disjoint union of balls.) @@ -309,7 +309,7 @@ The image of $\psi$ is the subcomplex $G_*\sub \bc(X)$ generated by blob diagrams which split over some decomposition of $J$. -It follows from Proposition \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to +It follows from Lemma \ref{thm:small-blobs} that $\bc_*(X)$ is homotopic to a subcomplex of $G_*$. Next we define a map $\phi:G_*\to \cT$ using the method of acyclic models.