%!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 sort-of-simplicial set.

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

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}.

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 $\bc_*(X)$.

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

\end{lemma}

\begin{proof}

It suffices to show that for any finitely generated pair of subcomplexes 

\[

	(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

\[

\]

for all $x\in C_*$.

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

$\sbc_0(X) = \bc_0$.

(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 $s(b)\in \sbc_1$ 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 term of a decomposition

of $X$ where $B$ is embedded.

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

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

$h_1(b) \in \bc_1(X)$ 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 

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 $\bar{f}_j:B\cup\text{collar}\to B$ 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 $\cV_1$

(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 $h_1(b) \in \bc_1(X)$ 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 $s(b)\in \sbc_2$ 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 \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

also satisfying conditions specified below.

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

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 \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.

\nn{should maybe have figure}

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 $\cV_r = \cU$.

\nn{should probably be more specific at the end}

\end{proof}

\medskip

Next we define the sort-of-simplicial 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.

\end{itemize}

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.

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 $h: \btc_*(B^n)\to \btc_*(B^n)$.

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 $r = s\circ q$.

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.

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

Let $c(r(y))\in \btc_{0,j+1}$ be the constant map from the cone of $P$ to $\BD_0$ taking

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)) .

\]

\nn{up to sign, at least}

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

\nn{ignoring signs}

\begin{align*}

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

			&= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x) + e(\bd_t x) \\

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

			&= x .

\end{align*}

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

\nn{ignoring signs}

\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) \quad\quad\text{(since $r(\bd_b x) = 0$)} \\

			&= x .

\end{align*}

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

\nn{ignoring signs}

\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*}

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

\nn{ignoring signs}

\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).

\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 Eilenber-Zilber theorem and the fact that

\[

	\BD_k(X\du Y) \cong \coprod_{i+j=k} \BD_i(X)\times\BD_j(Y) .

\]

\end{proof}

For $S\sub X$, we say that $a\in \btc_k(X)$ is {\it supported on $S$}

\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 that are 

small with respect to $\cU$.

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 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 \ref{lem:bc-btc}]

Armed with the above lemmas, we can now proceed similarly to the proof of \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 $h:C_*\to \btc_*(X)$ such that $h(D_*) \sub \bc_*(X)$

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

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

(by \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.

Since $\bd(b + h_1(\bd b)) = s(\bd b) \in \bc_2$, we can find

$s(b)\in \bc_2$ with $\bd(s(b)) = \bd(b + h_1(\bd b))$ (by \ref{disj-union-contract}).

By \ref{bt-contract} and \ref{btc-prod}, we can now find

$h_2(b) \in \btc_3$, 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 \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 $CH_*(X, Y)$ denote $C_*(\Homeo(X \to Y))$, 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 $CH_*(X) \deq CH_*(X, X)$.

(For convenience, we will permit the singular cells generating $CH_*(X, Y)$ to be more general

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

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

\eq{

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

}

well-defined up to homotopy,

such that

\begin{enumerate}

\item on $CH_0(X, Y) \otimes \bc_*(X)$ 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, 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 $\btc_*$.

And in fact for $\btc_*$ we get a sharper result: we can omit

the ``up to homotopy" qualifiers.

Let $f\in CH_k(X, Y)$, $f:P^k\to \Homeo(X \to Y)$ 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 $CH_0(X, Y)$ 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 $CH_*(X, Y)$ actions defined above are associative.

That is, the following diagram commutes up to homotopy:

\[ \xymatrix{

& CH_*(Y, Z) \ot \bc_*(Y) \ar[dr]^{e_{YZ}} & \\

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

& CH_*(X, Z) \ot \bc_*(X) \ar[ur]_{e_{XZ}} &

} \]

Here $\mu:CH_*(X, Y) \ot CH_*(Y, Z)\to CH_*(X, Z)$ is the map induced by composition

of homeomorphisms.

\end{thm}

\begin{proof}

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

\end{proof}

\noop{

\subsection{[older version still hanging around]}

\label{ss:old-evmap-remnants}

\nn{should comment at the start about any assumptions about smooth, PL etc.}