author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Thu, 18 Mar 2010 19:40:46 +0000 | |
changeset 221 | 77b0cdeb0fcd |
parent 141 | e1d24be683bb |
child 222 | 217b6a870532 |
permissions | -rw-r--r-- |
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%!TEX root = ../blob1.tex |
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\section{Basic properties of the blob complex} |
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\label{sec:basic-properties} |
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\begin{prop} \label{disjunion} |
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There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
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\end{prop} |
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\begin{proof} |
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Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
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(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
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blob diagram $(b_1, b_2)$ on $X \du Y$. |
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Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
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In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
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to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
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a pair of blob diagrams on $X$ and $Y$. |
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These two maps are compatible with our sign conventions. |
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The two maps are inverses of each other. |
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\nn{should probably say something about sign conventions for the differential |
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in a tensor product of chain complexes; ask Scott} |
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\end{proof} |
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For the next proposition we will temporarily restore $n$-manifold boundary |
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conditions to the notation. |
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Suppose that for all $c \in \cC(\bd B^n)$ |
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we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
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of the quotient map |
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$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
141 | 30 |
For example, this is always the case if the coefficient ring is a field. |
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Then |
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\begin{prop} \label{bcontract} |
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For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
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is a chain homotopy equivalence |
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with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
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Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
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\end{prop} |
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\begin{proof} |
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By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
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$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
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For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
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an $(i{+}1)$-st blob equal to all of $B^n$. |
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In other words, add a new outermost blob which encloses all of the others. |
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Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
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the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
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\end{proof} |
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Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
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equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
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For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
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where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
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221 | 54 |
\begin{cor} \label{disj-union-contract} |
55 |
If $X$ is a disjoint union of $n$-balls, then $\bc_*(X; c)$ is contractible. |
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\end{cor} |
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57 |
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\begin{proof} |
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This follows from \ref{disjunion} and \ref{bcontract}. |
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\end{proof} |
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Define the {\it support} of a blob diagram to be the union of all the |
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blobs of the diagram. |
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Define the support of a linear combination of blob diagrams to be the union of the |
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supports of the constituent diagrams. |
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For future use we prove the following lemma. |
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\begin{lemma} \label{support-shrink} |
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Let $L_* \sub \bc_*(X)$ be a subcomplex generated by some |
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subset of the blob diagrams on $X$, and let $f: L_* \to L_*$ |
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be a chain map which does not increase supports and which induces an isomorphism on |
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$H_0(L_*)$. |
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Then $f$ is homotopic (in $\bc_*(X)$) to the identity $L_*\to L_*$. |
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\end{lemma} |
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\begin{proof} |
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We will use the method of acyclic models. |
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Let $b$ be a blob diagram of $L_*$, let $S\sub X$ be the support of $b$, and let |
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$r$ be the restriction of $b$ to $X\setminus S$. |
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Note that $S$ is a disjoint union of balls. |
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Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. |
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note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. |
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Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), |
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so $f$ and the identity map are homotopic. |
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\end{proof} |
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\medskip |
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\nn{Maybe there is no longer a need to repeat the next couple of props here, since we also state them in the introduction. |
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But I think it's worth saying that the Diff actions will be enhanced later. |
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Maybe put that in the intro too.} |
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93 |
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As we noted above, |
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\begin{prop} |
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There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
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\qed |
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\end{prop} |
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99 |
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\begin{prop} |
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For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
141 | 103 |
of $n$-manifolds and homeomorphisms to the category of chain complexes and |
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(chain map) isomorphisms. |
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\qed |
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\end{prop} |
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107 |
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In particular, |
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\begin{prop} \label{diff0prop} |
141 | 110 |
There is an action of $\Homeo(X)$ on $\bc_*(X)$. |
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\qed |
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112 |
\end{prop} |
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113 |
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The above will be greatly strengthened in Section \ref{sec:evaluation}. |
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115 |
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116 |
\medskip |
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117 |
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For the next proposition we will temporarily restore $n$-manifold boundary |
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conditions to the notation. |
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120 |
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Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
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Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
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with boundary $Z\sgl$. |
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Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
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we have the blob complex $\bc_*(X; a, b, c)$. |
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If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
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$X$ to get blob diagrams on $X\sgl$: |
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|
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\begin{prop} |
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There is a natural chain map |
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\eq{ |
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\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
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} |
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The sum is over all fields $a$ on $Y$ compatible at their |
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($n{-}2$-dimensional) boundaries with $c$. |
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`Natural' means natural with respect to the actions of diffeomorphisms. |
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\qed |
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\end{prop} |
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The above map is very far from being an isomorphism, even on homology. |
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This will be fixed in Section \ref{sec:gluing} below. |
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%\nn{Next para not needed, since we already use bullet = gluing notation above(?)} |
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%An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
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%and $X\sgl = X_1 \cup_Y X_2$. |
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%(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
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%For $x_i \in \bc_*(X_i)$, we introduce the notation |
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%\eq{ |
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% x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
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%} |
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%Note that we have resumed our habit of omitting boundary labels from the notation. |
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