%!TEX root = ../blob1.tex

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

\label{sec:evaluation}

\nn{new plan: use the sort-of-simplicial space version of

the blob complex.

first define it, then show it's hty equivalent to the other def, then observe that

$CH*$ acts.

maybe salvage some of the original version of this section as a subsection outlining

how one might proceed directly.}

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 be fixed on the boundaries.)

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}.

For technical reasons we also show that requiring the blobs to be

embedded yields a homotopy equivalent complex.

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.

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 $p,p'\in P$.

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 $x\in M\setmin S$.

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{thm}[Small blobs] \label{thm:small-blobs-xx}

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

\end{thm}

\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 $x + h\bd(x) + \bd h(X) \in \sbc_*(X)_*(X)$ for all $x\in C_*$.

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

$\sbc_0(X) = \bc_0$.

(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 sufficiently small ball.

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 $f_j:\bc_0(B)\to\bc_0(B)$ 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 $\cV_1$.

\nn{need to say this better; maybe give fig}

Let $g_j:B\to B$ be the embedding at the $j$-th stage.

There are 1-blob diagrams $c_{ij} \in \bc_1(B)$ such that $c_{ij}$ is compatible with $\cV_1$

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

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.

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 $\bd(d_j) = g_j(s(\bd b)) - g_{j-1}(s(\bd b))$.

Let $g_{j-1}(s(\bd b)) = \sum e_k$, 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

Furthermore, we can arrange that all of the $q_m$ have the same support, and that this support

is contained in a open set of $\cV_1$.

(This is possible since there are only finitely many $p_m$.)

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

\label{ss:emap-def}

\subsection{[older version still hanging around]}

\label{ss:old-evmap-remnants}

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

\nn{should maybe mention alternate def of blob complex (sort-of-simplicial space instead of

sort-of-simplicial set) where this action would be easy}

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

than simplices --- they can be based on any linear polyhedron.

\nn{be more restrictive here?  does more need to be said?})

\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)

}

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}

Moreover, for any $m \geq 0$, we can find a family of chain maps $\{e_{XY}\}$ 

satisfying the above two conditions which is $m$-connected. In particular, this means that the choice of chain map above is unique up to homotopy.

\end{thm}

\begin{rem}

Note that the statement doesn't quite give uniqueness up to iterated homotopy. We fully expect that this should actually be the case, but haven't been able to prove this.

\end{rem}

Before giving the proof, we state the essential technical tool of Lemma \ref{extension_lemma}, 

and then give an outline of the method of proof.

Without loss of generality, we will assume $X = Y$.

\medskip

Let $f: P \times X \to X$ be a family of homeomorphisms (e.g. a generator of $CH_*(X)$)

and let $S \sub X$.

We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all

$x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if 

there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background"

homeomorphism $f_0 : X \to X$ so that

\begin{align*}

	f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\

\intertext{and}

	f(p,x) & = f_0(x) \;\;\;\; \mbox{for}\; (p, x) \in {P \times (X \setmin S)}.

\end{align*}

Note that if $f$ is supported on $S$ then it is also supported on any $R \sup S$.

(So when we talk about ``the" support of a family, there is some ambiguity,

but this ambiguity will not matter to us.)

Let $\cU = \{U_\alpha\}$ be an open cover of $X$.

A $k$-parameter family of homeomorphisms $f: P \times X \to X$ is

{\it adapted to $\cU$} 

if the support of $f$ is contained in the union

of at most $k$ of the $U_\alpha$'s.

\begin{lemma}  \label{extension_lemma}

Let $x \in CH_k(X)$ be a singular chain such that $\bd x$ is adapted to $\cU$.

Then $x$ is homotopic (rel boundary) to some $x' \in CH_k(X)$ which is adapted to $\cU$.

Furthermore, one can choose the homotopy so that its support is equal to the support of $x$.

\end{lemma}

The proof will be given in \S\ref{sec:localising}.

\medskip

Before diving into the details, we outline our strategy for the proof of Theorem \ref{thm:CH}.

Let $p$ be a singular cell in $CH_k(X)$ and $b$ be a blob diagram in $\bc_*(X)$.

We say that $p\ot b$ is {\it localizable} if there exists $V \sub X$ such that

\begin{itemize}

\item $V$ is homeomorphic to a disjoint union of balls, and

\item $\supp(p) \cup \supp(b) \sub V$.

\end{itemize}

(Recall that $\supp(b)$ is defined to be the union of the blobs of the diagram $b$.)

Assuming that $p\ot b$ is localizable as above, 

let $W = X \setmin V$, $W' = p(W)$ and $V' = X\setmin W'$.

We then have a factorization 

\[

	p = \gl(q, r),

\]

where $q \in CH_k(V, V')$ and $r \in CH_0(W, W')$.

We can also factorize $b = \gl(b_V, b_W)$, where $b_V\in \bc_*(V)$ and $b_W\in\bc_0(W)$.

According to the commutative diagram of the proposition, we must have

\[

	e_X(p\otimes b) = e_X(\gl(q\otimes b_V, r\otimes b_W)) = 

				gl(e_{VV'}(q\otimes b_V), e_{WW'}(r\otimes b_W)) .

\]

Since $r$ is a  0-parameter family of homeomorphisms, we must have

\[

	e_{WW'}(r\otimes b_W) = r(b_W),

\]

where $r(b_W)$ denotes the obvious action of homeomorphisms on blob diagrams (in

this case a 0-blob diagram).

Since $V'$ is a disjoint union of balls, $\bc_*(V')$ is acyclic in degrees $>0$ 

(by Properties \ref{property:disjoint-union} and \ref{property:contractibility}).

Assuming inductively that we have already defined $e_{VV'}(\bd(q\otimes b_V))$,

there is, up to (iterated) homotopy, a unique choice for $e_{VV'}(q\otimes b_V)$

such that 

\[

	\bd(e_{VV'}(q\otimes b_V)) = e_{VV'}(\bd(q\otimes b_V)) .

\]

Thus the conditions of the proposition determine (up to homotopy) the evaluation

map for localizable generators $p\otimes b$.

On the other hand, Lemma \ref{extension_lemma} allows us to homotope 

arbitrary generators to sums of localizable generators.

This (roughly) establishes the uniqueness part of the proposition.

To show existence, we must show that the various choices involved in constructing

evaluation maps in this way affect the final answer only by a homotopy.

Now for a little more detail.

(But we're still just motivating the full, gory details, which will follow.)

Choose a metric on $X$, and let $\cU_\gamma$ be the open cover of $X$ by balls of radius $\gamma$.

By Lemma \ref{extension_lemma} we can restrict our attention to $k$-parameter families 

$p$ of homeomorphisms such that $\supp(p)$ is contained in the union of $k$ $\gamma$-balls.

For fixed blob diagram $b$ and fixed $k$, it's not hard to show that for $\gamma$ small enough

$p\ot b$ must be localizable.

On the other hand, for fixed $k$ and $\gamma$ there exist $p$ and $b$ such that $p\ot b$ is not localizable,

and for fixed $\gamma$ and $b$ there exist non-localizable $p\ot b$ for sufficiently large $k$.

Thus we will need to take an appropriate limit as $\gamma$ approaches zero.

The construction of $e_X$, as outlined above, depends on various choices, one of which 

is the choice, for each localizable generator $p\ot b$, 

of disjoint balls $V$ containing $\supp(p)\cup\supp(b)$.

Let $V'$ be another disjoint union of balls containing $\supp(p)\cup\supp(b)$,

and assume that there exists yet another disjoint union of balls $W$ containing 

$V\cup V'$.

Then we can use $W$ to construct a homotopy between the two versions of $e_X$ 

associated to $V$ and $V'$.

If we impose no constraints on $V$ and $V'$ then such a $W$ need not exist.

Thus we will insist below that $V$ (and $V'$) be contained in small metric neighborhoods

of $\supp(p)\cup\supp(b)$.

Because we want not mere homotopy uniqueness but iterated homotopy uniqueness,

we will similarly require that $W$ be contained in a slightly larger metric neighborhood of 

$\supp(p)\cup\supp(b)$, and so on.

\begin{proof}[Proof of Theorem \ref{thm:CH}.]

We'll use the notation $|b| = \supp(b)$ and $|p| = \supp(p)$.

Choose a metric on $X$.

Choose a monotone decreasing sequence of positive real numbers $\ep_i$ converging to zero

(e.g.\ $\ep_i = 2^{-i}$).

Choose another sequence of positive real numbers $\delta_i$ such that $\delta_i/\ep_i$

converges monotonically to zero (e.g.\ $\delta_i = \ep_i^2$).

Let $\phi_l$ be an increasing sequence of positive numbers

satisfying the inequalities of Lemma \ref{xx2phi} below.

Given a generator $p\otimes b$ of $CH_*(X)\otimes \bc_*(X)$ and non-negative integers $i$ and $l$

define

\[

	N_{i,l}(p\ot b) \deq \Nbd_{l\ep_i}(|b|) \cup \Nbd_{\phi_l\delta_i}(|p|).

\]

In other words, for each $i$

we use the metric to choose nested neighborhoods of $|b|\cup |p|$ (parameterized

by $l$), with $\ep_i$ controlling the size of the buffers around $|b|$ and $\delta_i$ controlling

the size of the buffers around $|p|$.

Next we define subcomplexes $G_*^{i,m} \sub CH_*(X)\otimes \bc_*(X)$.

Let $p\ot b$ be a generator of $CH_*(X)\otimes \bc_*(X)$ and let $k = \deg(p\ot b)

= \deg(p) + \deg(b)$.

We say $p\ot b$ is in $G_*^{i,m}$ exactly when either (a) $\deg(p) = 0$ or (b)

there exist codimension-zero submanifolds $V_0,\ldots,V_m \sub X$ such that each $V_j$

is homeomorphic to a disjoint union of balls and

\[

	N_{i,k}(p\ot b) \subeq V_0 \subeq N_{i,k+1}(p\ot b)

			\subeq V_1 \subeq \cdots \subeq V_m \subeq N_{i,k+m+1}(p\ot b) ,

\]

and further $\bd(p\ot b) \in G_*^{i,m}$.

We also require that $b$ is splitable (transverse) along the boundary of each $V_l$.

Note that $G_*^{i,m+1} \subeq G_*^{i,m}$.

As sketched above and explained in detail below, 

$G_*^{i,m}$ is a subcomplex where it is easy to define

the evaluation map.

The parameter $m$ controls the number of iterated homotopies we are able to construct

(see Lemma \ref{m_order_hty}).

The larger $i$ is (i.e.\ the smaller $\ep_i$ is), the better $G_*^{i,m}$ approximates all of

$CH_*(X)\ot \bc_*(X)$ (see Lemma \ref{Gim_approx}).

Next we define a chain map (dependent on some choices) $e_{i,m}: G_*^{i,m} \to \bc_*(X)$.

(When the domain is clear from context we will drop the subscripts and write

simply  $e: G_*^{i,m} \to \bc_*(X)$).

Let $p\ot b \in G_*^{i,m}$.

If $\deg(p) = 0$, define

\[

	e(p\ot b) = p(b) ,

\]

where $p(b)$ denotes the obvious action of the homeomorphism(s) $p$ on the blob diagram $b$.

For general $p\ot b$ ($\deg(p) \ge 1$) assume inductively that we have already defined

$e(p'\ot b')$ when $\deg(p') + \deg(b') < k = \deg(p) + \deg(b)$.

Choose $V = V_0$ as above so that 

\[

	N_{i,k}(p\ot b) \subeq V \subeq N_{i,k+1}(p\ot b) .

\]

Let $\bd(p\ot b) = \sum_j p_j\ot b_j$, and let $V^j$ be the choice of neighborhood

of $|p_j|\cup |b_j|$ made at the preceding stage of the induction.

For all $j$, 

\[

	V^j \subeq N_{i,k}(p_j\ot b_j) \subeq N_{i,k}(p\ot b) \subeq V .

\]

(The second inclusion uses the facts that $|p_j| \subeq |p|$ and $|b_j| \subeq |b|$.)

We therefore have splittings

\[

	p = p'\bullet p'' , \;\; b = b'\bullet b'' , \;\; e(\bd(p\ot b)) = f'\bullet f'' ,

\]

where $p' \in CH_*(V)$, $p'' \in CH_*(X\setmin V)$, 

$b' \in \bc_*(V)$, $b'' \in \bc_*(X\setmin V)$, 

$f' \in \bc_*(p(V))$, and $f'' \in \bc_*(p(X\setmin V))$.

(Note that since the family of homeomorphisms $p$ is constant (independent of parameters)

near $\bd V$, the expressions $p(V) \sub X$ and $p(X\setmin V) \sub X$ are

unambiguous.)

We have $\deg(p'') = 0$ and, inductively, $f'' = p''(b'')$.

%We also have that $\deg(b'') = 0 = \deg(p'')$.

Choose $x' \in \bc_*(p(V))$ such that $\bd x' = f'$.

This is possible by Properties \ref{property:disjoint-union} and \ref{property:contractibility}  and the fact that isotopic fields

differ by a local relation.

Finally, define

\[

	e(p\ot b) \deq x' \bullet p''(b'') .

\]

Note that above we are essentially using the method of acyclic models \nn{\S \ref{sec:moam}}.

For each generator $p\ot b$ we specify the acyclic (in positive degrees) 

target complex $\bc_*(p(V)) \bullet p''(b'')$.

The definition of $e: G_*^{i,m} \to \bc_*(X)$ depends on two sets of choices:

The choice of neighborhoods $V$ and the choice of inverse boundaries $x'$.

The next lemma shows that up to (iterated) homotopy $e$ is independent

of these choices.

(Note that independence of choices of $x'$ (for fixed choices of $V$)

is a standard result in the method of acyclic models.)

%\begin{lemma}

%Let $\tilde{e} :  G_*^{i,m} \to \bc_*(X)$ be a chain map constructed like $e$ above, but with

%different choices of $x'$ at each step.

%(Same choice of $V$ at each step.)

%Then $e$ and $\tilde{e}$ are homotopic via a homotopy in $\bc_*(p(V)) \bullet p''(b'')$.

%Any two choices of such a first-order homotopy are second-order homotopic, and so on, 

%to arbitrary order.

%\end{lemma}