%!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 $C_*(\Homeo(X, Y))$ is 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 $\bc_*(X)$, recall from \S \ref{sec:basic-properties} that 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 this it in turn 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 $x\in C_*$.

By the splittings axiom for fields, any field is splittable into small pieces.

It follows that $\sbc_0(X) = \bc_0(X)$.

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 

$g_{j-1}(|f_j|)$ is also contained in 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, $|c_{ij}| = g_{j-1}(|f_j|)$)

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}

\medskip

Next we define the cone-product space version of the blob complex, $\btc_*(X)$.

First we must specify a topology on the set of $k$-blob diagrams, $\BD_k$.

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 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 $\BD_0(B)$.

\end{itemize}

We can summarize the above by saying that in the typical continuous family

$P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map

$P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. 

(``Varying independently'' means that after pulling back via the family of homeomorphisms to the original twig blob, 

one sees a continuous family of labels.)

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, 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. 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

\begin{lemma} \label{lem:bc-btc}

The inclusion $\bc_*(X) \sub \btc_*(X)$ is a homotopy equivalence

\end{lemma}

Before giving the proof we need a few preliminary results.

\begin{lemma} \label{bt-contract}

$\btc_*(B^n)$ is contractible (acyclic in positive degrees).

\end{lemma}

\begin{proof}

We will construct a contracting homotopy $h: \btc_*(B^n)\to \btc_{*+1}(B^n)$.

We will assume a splitting $s:H_0(\btc_*(B^n))\to \btc_0(B^n)$

of the quotient map $q:\btc_0(B^n)\to H_0(\btc_*(B^n))$.

Let $\rho = s\circ q$.

For $x\in \btc_{ij}$ with $i\ge 1$ define

\[

	h(x) = e(x) ,

\]

where

\[

	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 $\rho\circ y : P\to \BD_0$. 

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

whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$.

Now define, for $y\in \btc_{0j}$,

\[

	h(y) = e(y - r(y)) - c(r(y)) .

\]

We must now verify that $h$ does the job it was intended to do.

For $x\in \btc_{ij}$ with $i\ge 2$ we have

\begin{align*}

	\bd h(x) + h(\bd x) &= \bd(e(x)) + e(\bd x) && \\

			&= \bd_b(e(x)) + (-1)^{i+1} \bd_t(e(x)) + e(\bd_b x) + (-1)^i e(\bd_t x) && \\

			&= \bd_b(e(x)) + e(\bd_b x) && \text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\

		 	&= x . &&

\end{align*}

For $x\in \btc_{1j}$ we have

\begin{align*}

	\bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) - c(r(\bd_b x)) - e(\bd_t x) && \\

			&= \bd_b(e(x)) + e(\bd_b x) && \text{(since $r(\bd_b x) = 0$)} \\

			&= x . &&

\end{align*}

For $x\in \btc_{0j}$ with $j\ge 1$ we have

\begin{align*}

	\bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) + \bd_t(c(r(x))) + 

											e(\bd_t x - r(\bd_t x)) - c(r(\bd_t x)) \\

			&= x - r(x) + \bd_t(c(r(x))) - c(r(\bd_t x)) \\

			&= 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 $\bd_t(e(r(x))) = e(r(\bd_t x))$

and $\bd_t(c(r(x))) - c(r(\bd_t x))  = r(x)$.

For $x\in \btc_{00}$ we have

\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). \qedhere

\end{align*}

\end{proof}

\begin{lemma} \label{btc-prod}

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 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 $S$}

and $r\in \btc_0(X\setmin S)$ such that $a = a'\bullet r$.

\newcommand\sbtc{\btc^{\cU}}

Let $\cU$ be an open cover of $X$.

Let $\sbtc_*(X)\sub\btc_*(X)$ be the subcomplex generated by

$a\in \btc_*(X)$ such that there is a decomposition $X = \cup_i D_i$

such that each $D_i$ is a ball contained in some open set of $\cU$ and

$a$ is splittable along this decomposition.

In other words, $a$ can be obtained by gluing together pieces, each of which

is small with respect to $\cU$.

\begin{lemma} \label{small-top-blobs}

For any open cover $\cU$ of $X$, the inclusion $\sbtc_*(X)\sub\btc_*(X)$

is a homotopy equivalence.

\end{lemma}

\begin{proof}

This follows from a combination of Lemma \ref{extension_lemma_c} and the techniques of

the proof of Lemma \ref{small-blobs-b}.

It suffices to show that we can deform a finite subcomplex $C_*$ of $\btc_*(X)$ into $\sbtc_*(X)$

(relative to any designated subcomplex of $C_*$ already in $\sbtc_*(X)$).

The first step is to replace families of general blob diagrams with families 

of blob diagrams that are small with respect to $\cU$.

(If $f:P \to \BD_k$ is the family then for all $p\in P$ we have that $f(p)$ is a diagram in which the blobs are small.)

This is done as in the proof of Lemma \ref{small-blobs-b}; the technique of the proof works in families.

Each such family is homotopic to a sum of families which can be a ``lifted" to $\Homeo(X)$.

That is, $f:P \to \BD_k$ has the form $f(p) = g(p)(b)$ for some $g:P\to \Homeo(X)$ and $b\in \BD_k$.

(We are ignoring a complication related to twig blob labels, which might vary

independently of $g$, but this complication does not affect the conclusion we draw here.)

We now apply Lemma \ref{extension_lemma_c} to get families which are supported 

on balls $D_i$ contained in open sets of $\cU$.

\end{proof}

\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 $h:C_*\to \btc_{*+1}(X)$ such that $h(D_*) \sub \bc_{*+1}(X)$

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.

We choose $\cU$ fine enough so that each generator of $C_*$ is supported on a disjoint union of balls.

(This is possible since the original $C_*$ was finite and therefore had bounded dimension.)

Since $\bc_0(X) = \btc_0(X)$, we can take $h_0 = 0$.

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 Corollary \ref{disj-union-contract}), and also $h_1(b) \in \btc_2(X)$

such that $\bd (h_1(b)) = s(b) - b$

(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.

$s(b)\in \bc_2(X)$ 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 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 $m$.

The latter can be achieved with finer control over the various

choices of disjoint unions of balls in the above proofs, but we will not pursue this here.

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

\label{ss:emap-def}

Let  $C_*(\Homeo(X \to Y))$ denote the singular chain complex of

the space of homeomorphisms

between the $n$-manifolds $X$ and $Y$ 

(any given singular chain extends a fixed homeomorphism $\bd X \to \bd Y$).

We also will use the abbreviated notation $\CH{X} \deq \CH{X \to X}$.

(For convenience, we will permit the singular cells generating $\CH{X \to Y}$ to be more general

than simplices --- they can be based on any cone-product polyhedron (see Remark \ref{blobsset-remark}).)

\begin{thm}  \label{thm:CH} \label{thm:evaluation}%

For $n$-manifolds $X$ and $Y$ there is a chain map

\eq{

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

}

well-defined up to homotopy,

such that

\begin{enumerate}

\item on $C_0(\Homeo(X \to Y)) \otimes \bc_*(X)$ it agrees with the obvious action of 

$\Homeo(X, Y)$ on $\bc_*(X)$  described in Property \ref{property:functoriality}, and

\item for any compatible splittings $X\to X\sgl$ and $Y\to Y\sgl$, 

the following diagram commutes up to homotopy

\begin{equation*}

\xymatrix@C+2cm{

      \CH{X \to Y} \otimes \bc_*(X)

        \ar[r]_(.6){e_{XY}}  \ar[d]^{\gl \otimes \gl}   &

            \bc_*(Y)\ar[d]^{\gl} \\

     \CH{X\sgl, Y\sgl} \otimes \bc_*(X\sgl) \ar[r]_(.6){e_{X\sgl Y\sgl}}   & 	\bc_*(Y\sgl)  

}

\end{equation*}

\end{enumerate}

\end{thm}

\begin{proof}

In light of Lemma \ref{lem:bc-btc}, it suffices to prove the theorem with 

$\bc_*$ replaced by $\btc_*$.

In fact, for $\btc_*$ we get a sharper result: we can omit

the ``up to homotopy" qualifiers.

Let $f\in C_k(\Homeo(X \to Y))$, $f:P^k\to \Homeo(X \to Y)$ and $a\in \btc_{ij}(X)$, 

$a:Q^j \to \BD_i(X)$.

Define $e_{XY}(f\ot a)\in \btc_{i,j+k}(Y)$ by

\begin{align*}

	e_{XY}(f\ot a) : P\times Q &\to \BD_i(Y) \\

	(p,q) &\mapsto f(p)(a(q))  .

\end{align*}

It is clear that this agrees with the previously defined $C_0(\Homeo(X \to Y))$ action on $\btc_*$,

and it is also easy to see that the diagram in item 2 of the statement of the theorem

commutes on the nose.

\end{proof}

\begin{thm}

\label{thm:CH-associativity}

The $\CH{X \to Y}$ actions defined above are associative.

That is, the following diagram commutes up to homotopy:

\[ \xymatrix@C=5pt{

& \CH{Y\to Z} \ot \bc_*(Y) \ar[drr]^{e_{YZ}} & &\\

\CH{X \to Y} \ot \CH{Y \to Z} \ot \bc_*(X) \ar[ur]^{e_{XY}\ot\id} \ar[dr]_{\mu\ot\id} & & & \bc_*(Z) \\

& \CH{X \to Z} \ot \bc_*(X) \ar[urr]_{e_{XZ}} & &

} \]

Here $\mu:\CH{X\to Y} \ot \CH{Y \to Z}\to \CH{X \to Z}$ is the map induced by composition

of homeomorphisms.

\end{thm}

\begin{proof}

The corresponding diagram for $\btc_*$ commutes on the nose.

\end{proof}

\begin{remark} \label{collar-map-action-remark} \rm

Like $\Homeo(X)$, collar maps also have a natural topology (see discussion following Axiom \ref{axiom:families}),

and by adjusting the topology on blob diagrams we can arrange that families of collar maps

act naturally on $\btc_*(X)$.

\end{remark}