blob: comparison blob1.tex

equal deleted inserted replaced

-:af22fa790d13
+:15a79fb469e1
 \def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}}
 \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM;
 % \DeclareMathOperator{\pr}{pr} etc.
 \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}}
-\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp}{maps};
+\applytolist{declaremathop}{pr}{im}{gl}{ev}{coinv}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Maps}{Diff}{sign}{supp};
 %%%%%% end excerpt
 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 \nn{say more about this?} and
+These two maps are compatible with our sign conventions.
-with the linear label relations.
 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}
 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
 $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$.
-\nn{always the case if we're working over $\c$}.
+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 $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$.
 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 $x - s(p(x)) \in U(B^n; c)$.
-\nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$}
 \end{proof}
-(Note that for the above proof to work, we need the linear label relations
+Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
-for blob labels.
+equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.
-Also we need to blob reordering relations (?).)
+For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$,
-(Note also that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy
+where $(c', c'')$ is some (any) splitting of $c$ into domain and range.
-equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$.)
-(For fields based on $n$-cats, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$.)
 \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}
-% oops -- duplicate
-%\begin{prop} \label{functorialprop}
-%The assignment $X \mapsto \bc_*(X)$ extends to a functor from the category of
-%$n$-manifolds and homeomorphisms to the category of chain complexes and linear isomorphisms.
-%\end{prop}
-%\begin{proof}
-%Obvious.
-%\end{proof}
-%\nn{need to same something about boundaries and boundary conditions above.
-%maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
 \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}
-\nn{need to same something about boundaries and boundary conditions above.
-maybe fix the boundary and consider the category of $n$-manifolds with the given boundary.}
 In particular,
 \begin{prop}  \label{diff0prop}
 There is an action of $\Diff(X)$ on $\bc_*(X)$.
 \qed
 \end{prop}
 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 (pictures, boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$,
+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}
 \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.
-\bigskip
-\nn{what else?}
 \section{Hochschild homology when $n=1$}
 \label{sec:hochschild}
 \input{text/hochschild}

changeset 65	15a79fb469e1
parent 63	71b4e45f47f6
child 67	1df2e5b38eb2