diff -r 901a7c79976b -r ba4f86b15ff0 text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sat Jul 17 20:57:46 2010 -0600 +++ b/text/a_inf_blob.tex Sun Jul 18 08:07:50 2010 -0600 @@ -16,7 +16,10 @@ 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 $\cU$. +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. @@ -48,10 +51,11 @@ \[ \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . \] -In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ +On 0-simplices of the hocolimit +we just glue together the various blob diagrams on $X_i\times F$ (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on $Y\times F$. -In filtration degrees 1 and higher we define the map to be zero. +For simplices of dimension 1 and higher we define the map to be zero. It is easy to check that this is a chain map. In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ @@ -80,10 +84,10 @@ By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is $b$ split according to $K_0\times F$. To simplify notation we will just write plain $b$ instead of $b^\sharp$. -Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give -$a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, -filtration degree 2 stuff which kills the homology created by the -filtration degree 1 stuff, and so on. +Roughly speaking, $D(a)$ consists of 0-simplices which glue up to give +$a$ (or one of its iterated boundaries), 1-simplices which connect all the 0-simplices, +2-simplices which kill the homology created by the +1-simplices, and so on. More formally, \begin{lemma} \label{lem:d-a-acyclic} @@ -94,16 +98,15 @@ We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} leave the general case to the reader. -Let $K$ and $K'$ be two decompositions of $Y$ compatible with $a$. -We want to show that $(a, K)$ and $(a, K')$ are homologous via filtration degree 1 stuff. -\nn{need to say this better; these two chains don't have the same boundary.} +Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. +We want to find 1-simplices which connect $K$ and $K'$. We might hope that $K$ and $K'$ have a common refinement, but this is not necessarily the case. (Consider the $x$-axis and the graph of $y = x^2\sin(1/x)$ in $\r^2$.) However, we {\it can} find another decomposition $L$ such that $L$ shares common refinements with both $K$ and $K'$. Let $KL$ and $K'L$ denote these two refinements. -Then filtration degree 1 chains associated to the four anti-refinements +Then 1-simplices associated to the four anti-refinements $KL\to K$, $KL\to L$, $K'L\to L$ and $K'L\to K'$ give the desired chain connecting $(a, K)$ and $(a, K')$ (see Figure \ref{zzz4}). @@ -126,13 +129,13 @@ \end{figure} Consider a different choice of decomposition $L'$ in place of $L$ above. -This leads to a cycle consisting of filtration degree 1 stuff. -We want to show that this cycle bounds a chain of filtration degree 2 stuff. +This leads to a cycle of 1-simplices. +We want to find 2-simplices which fill in this cycle. Choose a decomposition $M$ which has common refinements with each of $K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$. (We also also require that $KLM$ antirefines to $KM$, etc.) -Then we have a filtration degree 2 chain, as shown in Figure \ref{zzz5}, which does the trick. -(Each small triangle in Figure \ref{zzz5} can be filled with a filtration degree 2 chain.) +Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick. +(Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.) \begin{figure}[!ht] %\begin{equation*} @@ -179,8 +182,8 @@ We are now in a position to apply the method of acyclic models to get a map $\phi:G_* \to \cl{\cC_F}(Y)$. -We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero -and $r$ has filtration degree greater than zero. +We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is a 0-simplex +and $r$ is a sum of simplices of dimension 1 or higher. We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. @@ -190,7 +193,7 @@ \] Roughly speaking, $(a, K)$ is just $a$ chopped up into little pieces, and $\psi$ glues those pieces back together, yielding $a$. -We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. +We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices. Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.