%!TEX root = ../blob1.tex

\section{Basic properties of the blob complex}

\label{sec:basic-properties}

\begin{prop} \label{disjunion}

There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$.

\end{prop}

\begin{proof}

Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them

(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a

blob diagram $(b_1, b_2)$ on $X \du Y$.

Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way.

In the other direction, any blob diagram on $X\du Y$ is equal (up to sign)

to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines

a pair of blob diagrams on $X$ and $Y$.

These two maps are compatible with our sign conventions.

The two maps are inverses of each other.

\nn{should probably say something about sign conventions for the differential

in a tensor product of chain complexes; ask Scott}

\end{proof}

For the next proposition we will temporarily restore $n$-manifold boundary

conditions to the notation.

Suppose that for all $c \in \cC(\bd B^n)$

we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$

of the quotient map

$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.

For example, this is always the case if you coefficient ring is a field.

Then

\begin{prop} \label{bcontract}

For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$

is a chain homotopy equivalence

with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.

Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0.

\end{prop}

\begin{proof}

By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map

$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$.

For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding

an $(i{+}1)$-st blob equal to all of $B^n$.

In other words, add a new outermost blob which encloses all of the others.

Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to

the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$.

\end{proof}

Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy

equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.

For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$,

where $(c', c'')$ is some (any) splitting of $c$ into domain and range.

\medskip

\nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction.

But I think it's worth saying that the Diff actions will be enhanced later.

Maybe put that in the intro too.}

As we noted above,

\begin{prop}

There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$.

\qed

\end{prop}

\begin{prop}

For fixed fields ($n$-cat), $\bc_*$ is a functor from the category

of $n$-manifolds and diffeomorphisms to the category of chain complexes and

(chain map) isomorphisms.

\qed

\end{prop}

In particular,

\begin{prop}  \label{diff0prop}

There is an action of $\Diff(X)$ on $\bc_*(X)$.

\qed

\end{prop}

The above will be greatly strengthened in Section \ref{sec:evaluation}.

\medskip

For the next proposition we will temporarily restore $n$-manifold boundary

conditions to the notation.

Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$.

Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$

with boundary $Z\sgl$.

Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,

we have the blob complex $\bc_*(X; a, b, c)$.

If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on

$X$ to get blob diagrams on $X\sgl$:

\begin{prop}

There is a natural chain map

\eq{

    \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl).

}

The sum is over all fields $a$ on $Y$ compatible at their

($n{-}2$-dimensional) boundaries with $c$.

`Natural' means natural with respect to the actions of diffeomorphisms.

\qed

\end{prop}

The above map is very far from being an isomorphism, even on homology.

This will be fixed in Section \ref{sec:gluing} below.

\nn{Next para not need, since we already use bullet = gluing notation above(?)}

An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$

and $X\sgl = X_1 \cup_Y X_2$.

(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.)

For $x_i \in \bc_*(X_i)$, we introduce the notation

\eq{

    x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) .

}

Note that we have resumed our habit of omitting boundary labels from the notation.