%!TEX root = ../blob1.tex

\section{Action of \texorpdfstring{$\CH{X}$}{C*(Homeo(M))}}

\label{sec:evaluation}

In this section we extend the action of homeomorphisms on $\bc_*(X)$

to an action of {\it families} of homeomorphisms.

That is, for each pair of homeomorphic manifolds $X$ and $Y$

we define a chain map

\[

    e_{XY} : CH_*(X, Y) \otimes \bc_*(X) \to \bc_*(Y) ,

\]

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 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, $\btc_*(X)$,

which is more amenable to this sort of action.

Recall from Remark \ref{blobsset-remark} that blob diagrams

have the structure of a cone-product set.

Blob diagrams can also be equipped with a natural topology, which converts this

cone-product set into a cone-product space.

Taking singular chains of this space we get $\btc_*(X)$.

The details are in \S \ref{ss:alt-def}.

We also prove a useful result (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 $\bc_*(X)$.

%This was our original approach, but working out the details created a nearly unreadable mess.

%We have salvaged a sketch of that approach in \S \ref{ss:old-evmap-remnants}.

%

%\nn{should revisit above intro after this section is done}

\subsection{Alternative definitions of the blob complex}

\label{ss:alt-def}

\newcommand\sbc{\bc^{\cU}}

In this subsection we define a subcomplex (small blobs) and supercomplex (families of blobs)

of the blob complex, and show that they are both homotopy equivalent to $\bc_*(X)$.

\medskip

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.

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 $r\in \bc_0(X\setmin S)$.

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 $Y$.

If $f: M \cup (Y\times I) \to M$ is a collaring homeomorphism

(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" $\bc^{\cU}_*(X)$ 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 $i: \bc^{\cU}_*(X) \into \bc_*(X)$ is a homotopy equivalence.

\end{lemma}

\begin{proof}

Since both complexes are free, it suffices to show that the inclusion induces

an isomorphism of homotopy groups.

To show that it suffices 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

\[

	h\bd(x) + \bd h(x) + x \in \sbc_*(X)

\]

for all $x\in C_*$.

For simplicity we will assume that all fields are splittable into small pieces, so that

$\sbc_0(X) = \bc_0(X)$.

(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 $s(b)\in \sbc_1(X)$ 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 stage of a decomposition

of $X$ where $B$ is embedded.

See Definition \ref{defn:configuration} and preceding discussion.)

It then follows from Corollary \ref{disj-union-contract} that we can choose

$h_1(b) \in \bc_2(X)$ such that $\bd(h_1(b)) = s(b) - b$.

Roughly speaking, $s(b)$ consists of a series of 1-blob diagrams implementing a series

of small collar maps, plus a shrunken version of $b$.

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 

fine enough that a condition stated later in the proof is satisfied.

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 

specified at the end of this paragraph.

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 $\cV_1$

(more specifically, $|c_{ij}| = g_{j-1}(B)$)

and $\bd c_{ij} = g_{j-1}(a_i) - g_{j}(a_i)$.

Define

\[

	s(b) = \sum_{i,j} c_{ij} + g(b)

\]

and choose $h_1(b) \in \bc_2(X)$ such that 

\[

	\bd(h_1(b)) = s(b) - b .

\]

Next we define $h_2$.

Let $b\in C_2$ be a 2-blob diagram.

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 $s(b)\in \sbc_2(X)$ 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 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, 

$s(b)$ consists of a series of 2-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 

disjoint union of balls.

Let $\cV_2$ be an auxiliary open cover of $X$, subordinate to $\cU$ and 

fine enough that a condition stated later in the proof is satisfied.

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.

Fix $j$.

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

$\bd q_m = g_{j-1}(p_m) - g_j(p_m)$ 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 $k$, $g_{j-1}(e_k) - q(\bd e_k)$.

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 $f_j$.

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 $U$

such that each individual ball lies in an open set of $\cV_2$.

(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 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 $d_j \deq \sum x_k$.

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.

We now define 

\[

	s(b) = \sum d_j + g(b),

\]

where $g$ is the composition of all the $f_j$'s.

It is easy to verify that $s(b) \in \sbc_2$, $\supp(s(b)) = \supp(b)$, and 

$\bd(s(b)) = s(\bd b)$.

If follows that we can choose $h_2(b)\in \bc_2(X)$ such that $\bd(h_2(b)) = s(b) - b - h_1(\bd b)$.

This completes the definition of $h_2$.

The general case $h_l$ is similar.

When constructing the analogue of $x_k$ above, we will need to find a disjoint union of balls $U$

which contains finitely many open sets from $\cV_{l-1}$

such that each ball is contained in some open set of $\cV_l$.

For sufficiently fine $\cV_{l-1}$ this will be possible.

Since $C_*$ is finite, the process terminates after finitely many, say $r$, steps.

We take $\cV_r = \cU$.

\end{proof}