author | Kevin Walker <kevin@canyon23.net> |
Wed, 07 Jul 2010 11:07:48 -0600 | |
changeset 422 | d55b85632926 |
parent 417 | d3b05641e7ca |
child 426 | 8aca80203f9d |
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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In this section we complete the proofs of Properties 2-4. |
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Throughout the paper, where possible, we prove results using Properties 1-4, |
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rather than the actual definition of blob homology. |
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This allows the possibility of future improvements to or alternatives on our definition. |
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In fact, we hope that there may be a characterisation of blob homology in |
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terms of Properties 1-4, but at this point we are unaware of one. |
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Recall Property \ref{property:disjoint-union}, |
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that there is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
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\begin{proof}[Proof of Property \ref{property:disjoint-union}] |
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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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(We follow the usual convention for tensors products of complexes, |
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as in e.g. \cite{MR1438306}: $d(a \tensor b) = da \tensor b + (-1)^{\deg(a)} a \tensor db$.) |
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The two maps are inverses of each other. |
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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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$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
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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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This proves Property \ref{property:contractibility} (the second half of the |
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statement of this Property was immediate from the definitions). |
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Note that even when 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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\begin{cor} \label{disj-union-contract} |
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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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\begin{proof} |
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This follows from Properties \ref{property:disjoint-union} and \ref{property:contractibility}. |
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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. \nn{We should actually have a section with a definition of ``compatible" and this statement as a lemma} |
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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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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 $b = a$, then we can glue up blob diagrams on |
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$X$ to get blob diagrams on $X\sgl$. |
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This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
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\textbf{Property \ref{property:gluing-map}.}\emph{ |
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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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} |
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This map is very far from being an isomorphism, even on homology. |
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We fix this deficit in Section \ref{sec:gluing} below. |