author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Tue, 21 Jul 2009 16:21:20 +0000 | |
changeset 98 | ec3af8dfcb3c |
parent 94 | 38ceade5cc5d |
child 100 | c5a43be00ed4 |
permissions | -rw-r--r-- |
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\documentclass[11pt,leqno]{amsart} |
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\newcommand{\pathtotrunk}{./} |
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\input{text/article_preamble.tex} |
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\input{text/top_matter.tex} |
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\input{text/kw_macros.tex} |
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\title{Blob Homology} |
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\begin{document} |
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\makeatletter |
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\@addtoreset{equation}{section} |
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\gdef\theequation{\thesection.\arabic{equation}} |
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\makeatother |
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\maketitle |
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\textbf{Draft version, do not distribute.} |
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%\versioninfo |
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[later than 11 June 2009] |
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\noop{ |
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\section*{Todo} |
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\subsection*{What else?...} |
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\begin{itemize} |
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\item higher priority |
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\begin{itemize} |
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\item K\&S: learn the state of the art in A-inf categories |
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(tensor products, Kadeishvili result, ...) |
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\item K: so-called evaluation map stuff |
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\item K: topological fields |
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\item section describing intended applications |
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\item say something about starting with semisimple n-cat (trivial?? not trivial?) |
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\item T.O.C. |
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\end{itemize} |
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\item medium priority |
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\begin{itemize} |
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\item $n=2$ examples |
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\item dimension $n+1$ (generalized Deligne conjecture?) |
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\item should be clear about PL vs Diff; probably PL is better |
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(or maybe not) |
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\item something about higher derived coend things (derived 2-coend, e.g.) |
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\item shuffle product vs gluing product (?) |
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\item connection between $A_\infty$ operad and topological $A_\infty$ cat defs |
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\end{itemize} |
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\item lower priority |
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\begin{itemize} |
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\item Derive Hochschild standard results from blob point of view? |
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\item Kh |
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\item Mention somewhere \cite{MR1624157} ``Skein homology''; it's not directly related, but has similar motivations. |
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\end{itemize} |
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\end{itemize} |
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} %end \noop |
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\input{text/intro.tex} |
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\section{Definitions} |
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\label{sec:definitions} |
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\subsection{Systems of fields} |
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\label{sec:fields} |
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Let $\cM_k$ denote the category (groupoid, in fact) with objects |
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oriented PL manifolds of dimension |
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$k$ and morphisms homeomorphisms. |
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(We could equally well work with a different category of manifolds --- |
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unoriented, topological, smooth, spin, etc. --- but for definiteness we |
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will stick with oriented PL.) |
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Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
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A $n$-dimensional {\it system of fields} in $\cS$ |
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is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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together with some additional data and satisfying some additional conditions, all specified below. |
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\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
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Before finishing the definition of fields, we give two motivating examples |
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(actually, families of examples) of systems of fields. |
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The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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from X to $B$. |
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The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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$j$-morphisms of $C$. |
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One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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This is described in more detail below. |
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Now for the rest of the definition of system of fields. |
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\begin{enumerate} |
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\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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and these maps are a natural |
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transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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$\cC(X)$ which restricts to $c$. |
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In this context, we will call $c$ a boundary condition. |
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\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
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\item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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again comprise a natural transformation of functors. |
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In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
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\item $\cC_k$ is compatible with the symmetric monoidal |
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structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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restriction maps. |
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\item Gluing without corners. |
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Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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copies of $Y$ in $\bd X$. |
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Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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|
128 |
Then (here's the axiom/definition part) there is an injective ``gluing" map |
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129 |
\[ |
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130 |
\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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131 |
\] |
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132 |
and this gluing map is compatible with all of the above structure (actions |
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|
133 |
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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|
134 |
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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|
135 |
the gluing map is surjective. |
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|
136 |
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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|
137 |
gluing surface, we say that fields in the image of the gluing map |
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|
138 |
are transverse to $Y$ or cuttable along $Y$. |
60
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139 |
\item Gluing with corners. |
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140 |
Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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|
141 |
Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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142 |
Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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|
143 |
(without corners) along two copies of $\bd Y$. |
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144 |
Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
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145 |
$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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|
146 |
Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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|
147 |
(This restriction map uses the gluing without corners map above.) |
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148 |
Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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149 |
maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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150 |
copies of $Y$ in $\bd X$. |
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151 |
Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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152 |
Then (here's the axiom/definition part) there is an injective ``gluing" map |
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|
153 |
\[ |
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154 |
\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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155 |
\] |
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156 |
and this gluing map is compatible with all of the above structure (actions |
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|
157 |
of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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158 |
Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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159 |
the gluing map is surjective. |
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|
160 |
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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161 |
gluing surface, we say that fields in the image of the gluing map |
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162 |
are transverse to $Y$ or cuttable along $Y$. |
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|
163 |
\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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164 |
$c \mapsto c\times I$. |
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165 |
These maps comprise a natural transformation of functors, and commute appropriately |
62
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166 |
with all the structure maps above (disjoint union, boundary restriction, etc.). |
60
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167 |
Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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|
168 |
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
59
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169 |
\end{enumerate} |
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170 |
|
62
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171 |
\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
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172 |
|
61
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|
173 |
\bigskip |
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|
174 |
Using the functoriality and $\bullet\times I$ properties above, together |
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|
175 |
with boundary collar homeomorphisms of manifolds, we can define the notion of |
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176 |
{\it extended isotopy}. |
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177 |
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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178 |
of $\bd M$. |
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179 |
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
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180 |
Let $c$ be $x$ restricted to $Y$. |
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|
181 |
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
62
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|
182 |
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
61
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183 |
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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|
184 |
Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
61
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|
185 |
More generally, we define extended isotopy to be the equivalence relation on fields |
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|
186 |
on $M$ generated by isotopy plus all instance of the above construction |
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187 |
(for all appropriate $Y$ and $x$). |
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|
188 |
|
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|
189 |
\nn{should also say something about pseudo-isotopy} |
59
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|
190 |
|
67 | 191 |
%\bigskip |
192 |
%\hrule |
|
193 |
%\bigskip |
|
194 |
% |
|
195 |
%\input{text/fields.tex} |
|
196 |
% |
|
197 |
% |
|
198 |
%\bigskip |
|
199 |
%\hrule |
|
200 |
%\bigskip |
|
60
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|
201 |
|
8 | 202 |
\nn{note: probably will suppress from notation the distinction |
0 | 203 |
between fields and their (orientation-reversal) duals} |
204 |
||
205 |
\nn{remark that if top dimensional fields are not already linear |
|
206 |
then we will soon linearize them(?)} |
|
207 |
||
60
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208 |
We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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|
209 |
by $n$-category morphisms. |
0 | 210 |
|
8 | 211 |
Given an $n$-category $C$ with the right sort of duality |
212 |
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
|
0 | 213 |
we can construct a system of fields as follows. |
214 |
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
|
215 |
with codimension $i$ cells labeled by $i$-morphisms of $C$. |
|
216 |
We'll spell this out for $n=1,2$ and then describe the general case. |
|
217 |
||
218 |
If $X$ has boundary, we require that the cell decompositions are in general |
|
219 |
position with respect to the boundary --- the boundary intersects each cell |
|
220 |
transversely, so cells meeting the boundary are mere half-cells. |
|
221 |
||
222 |
Put another way, the cell decompositions we consider are dual to standard cell |
|
223 |
decompositions of $X$. |
|
224 |
||
225 |
We will always assume that our $n$-categories have linear $n$-morphisms. |
|
226 |
||
227 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
|
228 |
an object (0-morphism) of the 1-category $C$. |
|
229 |
A field on a 1-manifold $S$ consists of |
|
230 |
\begin{itemize} |
|
8 | 231 |
\item A cell decomposition of $S$ (equivalently, a finite collection |
0 | 232 |
of points in the interior of $S$); |
8 | 233 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
0 | 234 |
by an object (0-morphism) of $C$; |
8 | 235 |
\item a transverse orientation of each 0-cell, thought of as a choice of |
0 | 236 |
``domain" and ``range" for the two adjacent 1-cells; and |
8 | 237 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
0 | 238 |
domain and range determined by the transverse orientation and the labelings of the 1-cells. |
239 |
\end{itemize} |
|
240 |
||
241 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
|
8 | 242 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
0 | 243 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
244 |
of the algebra. |
|
245 |
||
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246 |
\medskip |
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247 |
|
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248 |
For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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249 |
that are common in the literature. |
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250 |
We describe these carefully here. |
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251 |
|
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|
252 |
A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
0 | 253 |
an object of the 2-category $C$. |
254 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
|
255 |
A field on a 2-manifold $Y$ consists of |
|
256 |
\begin{itemize} |
|
8 | 257 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
0 | 258 |
that each component of the complement is homeomorphic to a disk); |
8 | 259 |
\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
0 | 260 |
by a 0-morphism of $C$; |
8 | 261 |
\item a transverse orientation of each 1-cell, thought of as a choice of |
0 | 262 |
``domain" and ``range" for the two adjacent 2-cells; |
8 | 263 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with |
264 |
domain and range determined by the transverse orientation of the 1-cell |
|
0 | 265 |
and the labelings of the 2-cells; |
8 | 266 |
\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
0 | 267 |
of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
268 |
to $\pm 1 \in S^1$; and |
|
8 | 269 |
\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
0 | 270 |
determined by the labelings of the 1-cells and the parameterizations of the previous |
271 |
bullet. |
|
272 |
\end{itemize} |
|
273 |
\nn{need to say this better; don't try to fit everything into the bulleted list} |
|
274 |
||
275 |
For general $n$, a field on a $k$-manifold $X^k$ consists of |
|
276 |
\begin{itemize} |
|
8 | 277 |
\item A cell decomposition of $X$; |
278 |
\item an explicit general position homeomorphism from the link of each $j$-cell |
|
0 | 279 |
to the boundary of the standard $(k-j)$-dimensional bihedron; and |
8 | 280 |
\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
0 | 281 |
domain and range determined by the labelings of the link of $j$-cell. |
282 |
\end{itemize} |
|
283 |
||
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|
284 |
%\nn{next definition might need some work; I think linearity relations should |
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%be treated differently (segregated) from other local relations, but I'm not sure |
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286 |
%the next definition is the best way to do it} |
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287 |
|
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288 |
\medskip |
0 | 289 |
|
8 | 290 |
For top dimensional ($n$-dimensional) manifolds, we're actually interested |
0 | 291 |
in the linearized space of fields. |
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292 |
By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
8 | 293 |
the vector space of finite |
0 | 294 |
linear combinations of fields on $X$. |
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295 |
If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
0 | 296 |
Thus the restriction (to boundary) maps are well defined because we never |
297 |
take linear combinations of fields with differing boundary conditions. |
|
298 |
||
299 |
In some cases we don't linearize the default way; instead we take the |
|
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300 |
spaces $\lf(X; a)$ to be part of the data for the system of fields. |
0 | 301 |
In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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302 |
Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
0 | 303 |
obvious relations on 0-cell labels. |
8 | 304 |
More specifically, let $L$ be a cell decomposition of $X$ |
0 | 305 |
and let $p$ be a 0-cell of $L$. |
306 |
Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
|
307 |
$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
|
308 |
Then the subspace $K$ is generated by things of the form |
|
309 |
$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
|
310 |
to infer the meaning of $\alpha_{\lambda c + d}$. |
|
311 |
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
|
312 |
||
8 | 313 |
\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
0 | 314 |
will do something similar below; in general, whenever a label lives in a linear |
8 | 315 |
space we do something like this; ? say something about tensor |
0 | 316 |
product of all the linear label spaces? Yes:} |
317 |
||
318 |
For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
|
319 |
Define an ``almost-field" to be a field without labels on the 0-cells. |
|
320 |
(Recall that 0-cells are labeled by $n$-morphisms.) |
|
321 |
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
|
322 |
space determined by the labeling of the link of the 0-cell. |
|
323 |
(If the 0-cell were labeled, the label would live in this space.) |
|
324 |
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
|
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325 |
We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
0 | 326 |
above tensor products. |
327 |
||
328 |
||
329 |
||
330 |
\subsection{Local relations} |
|
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\label{sec:local-relations} |
0 | 332 |
|
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333 |
|
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334 |
A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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335 |
for all $n$-manifolds $B$ which are |
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336 |
homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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337 |
satisfying the following properties. |
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338 |
\begin{enumerate} |
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339 |
\item functoriality: |
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340 |
$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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341 |
\item local relations imply extended isotopy: |
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342 |
if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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343 |
to $y$, then $x-y \in U(B; c)$. |
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344 |
\item ideal with respect to gluing: |
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345 |
if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
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346 |
\end{enumerate} |
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347 |
See \cite{kw:tqft} for details. |
0 | 348 |
|
349 |
||
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350 |
For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
0 | 351 |
where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
352 |
||
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353 |
For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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354 |
$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
0 | 355 |
domain and range. |
356 |
||
357 |
\nn{maybe examples of local relations before general def?} |
|
358 |
||
359 |
Given a system of fields and local relations, we define the skein space |
|
360 |
$A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
|
361 |
the $n$-manifold $Y$ modulo local relations. |
|
362 |
The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
|
363 |
is defined to be the dual of $A(Y; c)$. |
|
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364 |
(See \cite{kw:tqft} or xxxx for details.) |
0 | 365 |
|
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366 |
\nn{should expand above paragraph} |
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367 |
|
0 | 368 |
The blob complex is in some sense the derived version of $A(Y; c)$. |
369 |
||
370 |
||
371 |
||
372 |
\subsection{The blob complex} |
|
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373 |
\label{sec:blob-definition} |
0 | 374 |
|
375 |
Let $X$ be an $n$-manifold. |
|
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376 |
Assume a fixed system of fields and local relations. |
0 | 377 |
In this section we will usually suppress boundary conditions on $X$ from the notation |
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378 |
(e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
0 | 379 |
|
8 | 380 |
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
0 | 381 |
submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
382 |
$\overline{X \setmin Y}$. |
|
383 |
||
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|
384 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
0 | 385 |
|
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386 |
Define $\bc_0(X) = \lf(X)$. |
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|
387 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
0 | 388 |
We'll omit this sort of detail in the rest of this section.) |
389 |
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
|
390 |
||
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391 |
$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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392 |
Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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|
393 |
combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
0 | 394 |
\begin{itemize} |
395 |
\item An embedded closed ball (``blob") $B \sub X$. |
|
396 |
\item A field $r \in \cC(X \setmin B; c)$ |
|
397 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
|
398 |
\item A local relation field $u \in U(B; c)$ |
|
399 |
(same $c$ as previous bullet). |
|
400 |
\end{itemize} |
|
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|
401 |
In order to get the linear structure correct, we (officially) define |
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|
402 |
\[ |
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|
403 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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|
404 |
\] |
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|
405 |
The first direct sum is indexed by all blobs $B\subset X$, and the second |
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406 |
by all boundary conditions $c \in \cC(\bd B)$. |
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|
407 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
0 | 408 |
|
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|
409 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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|
410 |
\[ |
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|
411 |
(B, u, r) \mapsto u\bullet r, |
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|
412 |
\] |
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|
413 |
where $u\bullet r$ denotes the linear |
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|
414 |
combination of fields on $X$ obtained by gluing $u$ to $r$. |
8 | 415 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
0 | 416 |
just erasing the blob from the picture |
417 |
(but keeping the blob label $u$). |
|
418 |
||
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|
419 |
Note that the skein space $A(X)$ |
0 | 420 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
421 |
||
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|
422 |
$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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|
423 |
local relations encoded in $\bc_1(X)$. |
8 | 424 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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|
425 |
2-blob diagrams, of which there are two types, disjoint and nested. |
0 | 426 |
|
427 |
A disjoint 2-blob diagram consists of |
|
428 |
\begin{itemize} |
|
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|
429 |
\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
0 | 430 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
431 |
(where $c_i \in \cC(\bd B_i)$). |
|
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|
432 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
0 | 433 |
\end{itemize} |
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|
434 |
We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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|
435 |
reversing the order of the blobs changes the sign. |
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|
436 |
Define $\bd(B_0, B_1, u_0, u_1, r) = |
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|
437 |
(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
0 | 438 |
In other words, the boundary of a disjoint 2-blob diagram |
439 |
is the sum (with alternating signs) |
|
440 |
of the two ways of erasing one of the blobs. |
|
441 |
It's easy to check that $\bd^2 = 0$. |
|
442 |
||
443 |
A nested 2-blob diagram consists of |
|
444 |
\begin{itemize} |
|
445 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
|
446 |
\item A field $r \in \cC(X \setmin B_0; c_0)$ |
|
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|
447 |
(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
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|
448 |
\item A local relation field $u_0 \in U(B_0; c_0)$. |
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|
449 |
\end{itemize} |
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|
450 |
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
0 | 451 |
(for some $c_1 \in \cC(B_1)$) and |
452 |
$r' \in \cC(X \setmin B_1; c_1)$. |
|
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|
453 |
Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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|
454 |
Note that the requirement that |
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|
455 |
local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
0 | 456 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
457 |
sum of the two ways of erasing one of the blobs. |
|
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458 |
If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
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459 |
It is again easy to check that $\bd^2 = 0$. |
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|
460 |
|
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461 |
\nn{should draw figures for 1, 2 and $k$-blob diagrams} |
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|
462 |
|
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|
463 |
As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
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|
464 |
(officially) |
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|
465 |
\begin{eqnarray*} |
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|
466 |
\bc_2(X) & \deq & |
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|
467 |
\left( |
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|
468 |
\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
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|
469 |
U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
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|
470 |
\right) \\ |
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|
471 |
&& \bigoplus \left( |
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|
472 |
\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
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473 |
U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
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|
474 |
\right) . |
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|
475 |
\end{eqnarray*} |
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|
476 |
The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
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477 |
but we didn't feel like introducing a notation for that. |
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478 |
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
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|
479 |
(rather than a new, linearly independent 2-blob diagram). |
0 | 480 |
|
481 |
Now for the general case. |
|
482 |
A $k$-blob diagram consists of |
|
483 |
\begin{itemize} |
|
484 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
|
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485 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
0 | 486 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
487 |
(The case $B_i = B_j$ is allowed. |
|
488 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
|
489 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
|
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490 |
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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|
491 |
(These are implied by the data in the next bullets, so we usually |
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|
492 |
suppress them from the notation.) |
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493 |
$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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|
494 |
if the latter space is not empty. |
0 | 495 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
63
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496 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
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|
497 |
is determined by the $c_i$'s. |
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498 |
$r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
0 | 499 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
500 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
|
501 |
If $B_i = B_j$ then $u_i = u_j$. |
|
502 |
\end{itemize} |
|
503 |
||
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504 |
If two blob diagrams $D_1$ and $D_2$ |
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505 |
differ only by a reordering of the blobs, then we identify |
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506 |
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
0 | 507 |
|
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|
508 |
$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
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|
509 |
As before, the official definition is in terms of direct sums |
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|
510 |
of tensor products: |
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|
511 |
\[ |
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512 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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|
513 |
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
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|
514 |
\] |
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|
515 |
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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|
516 |
$\overline{c}$ runs over all boundary conditions, again as described above. |
67 | 517 |
$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. |
0 | 518 |
|
519 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
|
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|
520 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
0 | 521 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
522 |
If $B_j$ is not a twig blob, this involves only decrementing |
|
523 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
|
524 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
|
525 |
if removing $B_j$ creates new twig blobs. |
|
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|
526 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
0 | 527 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
528 |
Finally, define |
|
529 |
\eq{ |
|
8 | 530 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
0 | 531 |
} |
532 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
|
533 |
Thus we have a chain complex. |
|
534 |
||
535 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
|
536 |
||
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|
537 |
\nn{?? remark about dendroidal sets} |
0 | 538 |
|
539 |
||
540 |
||
541 |
\section{Basic properties of the blob complex} |
|
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|
542 |
\label{sec:basic-properties} |
0 | 543 |
|
544 |
\begin{prop} \label{disjunion} |
|
545 |
There is a natural isomorphism $\bc_*(X \du Y) \cong \bc_*(X) \otimes \bc_*(Y)$. |
|
546 |
\end{prop} |
|
547 |
\begin{proof} |
|
548 |
Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them |
|
8 | 549 |
(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a |
0 | 550 |
blob diagram $(b_1, b_2)$ on $X \du Y$. |
551 |
Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. |
|
552 |
In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) |
|
553 |
to one that puts $X$ blobs before $Y$ blobs in the ordering, and so determines |
|
554 |
a pair of blob diagrams on $X$ and $Y$. |
|
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|
555 |
These two maps are compatible with our sign conventions. |
0 | 556 |
The two maps are inverses of each other. |
557 |
\nn{should probably say something about sign conventions for the differential |
|
558 |
in a tensor product of chain complexes; ask Scott} |
|
559 |
\end{proof} |
|
560 |
||
561 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
562 |
conditions to the notation. |
|
563 |
||
8 | 564 |
Suppose that for all $c \in \cC(\bd B^n)$ |
565 |
we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ |
|
0 | 566 |
of the quotient map |
567 |
$p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. |
|
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|
568 |
For example, this is always the case if you coefficient ring is a field. |
0 | 569 |
Then |
570 |
\begin{prop} \label{bcontract} |
|
571 |
For all $c \in \cC(\bd B^n)$ the natural map $p: \bc_*(B^n, c) \to H_0(\bc_*(B^n, c))$ |
|
572 |
is a chain homotopy equivalence |
|
573 |
with inverse $s: H_0(\bc_*(B^n, c)) \to \bc_*(B^n; c)$. |
|
574 |
Here we think of $H_0(\bc_*(B^n, c))$ as a 1-step complex concentrated in degree 0. |
|
575 |
\end{prop} |
|
576 |
\begin{proof} |
|
577 |
By assumption $p\circ s = \id$, so all that remains is to find a degree 1 map |
|
578 |
$h : \bc_*(B^n; c) \to \bc_*(B^n; c)$ such that $\bd h + h\bd = \id - s \circ p$. |
|
579 |
For $i \ge 1$, define $h_i : \bc_i(B^n; c) \to \bc_{i+1}(B^n; c)$ by adding |
|
580 |
an $(i{+}1)$-st blob equal to all of $B^n$. |
|
581 |
In other words, add a new outermost blob which encloses all of the others. |
|
582 |
Define $h_0 : \bc_0(B^n; c) \to \bc_1(B^n; c)$ by setting $h_0(x)$ equal to |
|
583 |
the 1-blob with blob $B^n$ and label $x - s(p(x)) \in U(B^n; c)$. |
|
584 |
\end{proof} |
|
585 |
||
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586 |
Note that if there is no splitting $s$, we can let $h_0 = 0$ and get a homotopy |
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|
587 |
equivalence to the 2-step complex $U(B^n; c) \to \cC(B^n; c)$. |
0 | 588 |
|
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|
589 |
For fields based on $n$-categories, $H_0(\bc_*(B^n; c)) \cong \mor(c', c'')$, |
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|
590 |
where $(c', c'')$ is some (any) splitting of $c$ into domain and range. |
0 | 591 |
|
592 |
\medskip |
|
593 |
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|
594 |
\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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|
595 |
But I think it's worth saying that the Diff actions will be enhanced later. |
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|
596 |
Maybe put that in the intro too.} |
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|
597 |
|
0 | 598 |
As we noted above, |
599 |
\begin{prop} |
|
600 |
There is a natural isomorphism $H_0(\bc_*(X)) \cong A(X)$. |
|
601 |
\qed |
|
602 |
\end{prop} |
|
603 |
||
604 |
||
605 |
\begin{prop} |
|
606 |
For fixed fields ($n$-cat), $\bc_*$ is a functor from the category |
|
8 | 607 |
of $n$-manifolds and diffeomorphisms to the category of chain complexes and |
0 | 608 |
(chain map) isomorphisms. |
609 |
\qed |
|
610 |
\end{prop} |
|
611 |
||
612 |
In particular, |
|
613 |
\begin{prop} \label{diff0prop} |
|
614 |
There is an action of $\Diff(X)$ on $\bc_*(X)$. |
|
615 |
\qed |
|
616 |
\end{prop} |
|
617 |
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|
618 |
The above will be greatly strengthened in Section \ref{sec:evaluation}. |
0 | 619 |
|
620 |
\medskip |
|
621 |
||
622 |
For the next proposition we will temporarily restore $n$-manifold boundary |
|
623 |
conditions to the notation. |
|
624 |
||
625 |
Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
|
626 |
Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
|
627 |
with boundary $Z\sgl$. |
|
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|
628 |
Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
0 | 629 |
we have the blob complex $\bc_*(X; a, b, c)$. |
630 |
If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
|
631 |
$X$ to get blob diagrams on $X\sgl$: |
|
632 |
||
633 |
\begin{prop} |
|
634 |
There is a natural chain map |
|
635 |
\eq{ |
|
8 | 636 |
\gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
0 | 637 |
} |
8 | 638 |
The sum is over all fields $a$ on $Y$ compatible at their |
0 | 639 |
($n{-}2$-dimensional) boundaries with $c$. |
640 |
`Natural' means natural with respect to the actions of diffeomorphisms. |
|
641 |
\qed |
|
642 |
\end{prop} |
|
643 |
||
644 |
The above map is very far from being an isomorphism, even on homology. |
|
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|
645 |
This will be fixed in Section \ref{sec:gluing} below. |
0 | 646 |
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647 |
\nn{Next para not need, since we already use bullet = gluing notation above(?)} |
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changeset
|
648 |
|
0 | 649 |
An instance of gluing we will encounter frequently below is where $X = X_1 \du X_2$ |
650 |
and $X\sgl = X_1 \cup_Y X_2$. |
|
651 |
(Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) |
|
652 |
For $x_i \in \bc_*(X_i)$, we introduce the notation |
|
653 |
\eq{ |
|
8 | 654 |
x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . |
0 | 655 |
} |
656 |
Note that we have resumed our habit of omitting boundary labels from the notation. |
|
657 |
||
658 |
||
659 |
||
65
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|
660 |
|
0 | 661 |
|
15
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
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diff
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|
662 |
\section{Hochschild homology when $n=1$} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
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diff
changeset
|
663 |
\label{sec:hochschild} |
7340ab80db25
rearranging the Hochschild section. Splitting things up into lemmas, and explaining why those lemmas are what we need.
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parents:
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diff
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|
664 |
\input{text/hochschild} |
7
4ef2f77a4652
small to medium sized changes
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|
665 |
|
70 | 666 |
|
667 |
||
668 |
||
22
ada83e7228eb
rearranging; stating all the "properties" up front
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diff
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|
669 |
\section{Action of $\CD{X}$} |
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rearranging; stating all the "properties" up front
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diff
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|
670 |
\label{sec:evaluation} |
70 | 671 |
\input{text/evmap} |
69 | 672 |
|
673 |
||
70 | 674 |
|
94 | 675 |
\input{text/ncat.tex} |
18
aac9fd8d6bc6
finished with evaluation map stuff for now.
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17
diff
changeset
|
676 |
|
55 | 677 |
\input{text/A-infty.tex} |
34 | 678 |
|
55 | 679 |
\input{text/gluing.tex} |
51
195a0a91e062
continuing to write up commutative algebra stuff
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|
680 |
|
195a0a91e062
continuing to write up commutative algebra stuff
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50
diff
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|
681 |
|
195a0a91e062
continuing to write up commutative algebra stuff
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diff
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|
682 |
|
47
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|
683 |
\section{Commutative algebras as $n$-categories} |
939a4a5b1d80
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|
684 |
|
939a4a5b1d80
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|
685 |
\nn{this should probably not be a section by itself. i'm just trying to write down the outline |
939a4a5b1d80
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|
686 |
while it's still fresh in my mind.} |
939a4a5b1d80
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|
687 |
|
939a4a5b1d80
increased line width; start to write commutative algebra results
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|
688 |
If $C$ is a commutative algebra it |
48
b7ade62bea27
more commutative algebra stuff
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47
diff
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|
689 |
can (and will) also be thought of as an $n$-category whose $j$-morphisms are trivial for |
b7ade62bea27
more commutative algebra stuff
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|
690 |
$j<n$ and whose $n$-morphisms are $C$. |
47
939a4a5b1d80
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|
691 |
The goal of this \nn{subsection?} is to compute |
939a4a5b1d80
increased line width; start to write commutative algebra results
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changeset
|
692 |
$\bc_*(M^n, C)$ for various commutative algebras $C$. |
939a4a5b1d80
increased line width; start to write commutative algebra results
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changeset
|
693 |
|
939a4a5b1d80
increased line width; start to write commutative algebra results
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45
diff
changeset
|
694 |
Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
939a4a5b1d80
increased line width; start to write commutative algebra results
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changeset
|
695 |
|
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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changeset
|
696 |
Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
697 |
unlabeled points in $M$. |
48
b7ade62bea27
more commutative algebra stuff
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47
diff
changeset
|
698 |
Note that $\Sigma^0(M)$ is a point. |
47
939a4a5b1d80
increased line width; start to write commutative algebra results
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|
699 |
Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
939a4a5b1d80
increased line width; start to write commutative algebra results
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diff
changeset
|
700 |
|
49 | 701 |
Let $C_*(X, k)$ denote the singular chain complex of the space $X$ with coefficients in $k$. |
47
939a4a5b1d80
increased line width; start to write commutative algebra results
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changeset
|
702 |
|
51
195a0a91e062
continuing to write up commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
50
diff
changeset
|
703 |
\begin{prop} \label{sympowerprop} |
52 | 704 |
$\bc_*(M, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$. |
47
939a4a5b1d80
increased line width; start to write commutative algebra results
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45
diff
changeset
|
705 |
\end{prop} |
939a4a5b1d80
increased line width; start to write commutative algebra results
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45
diff
changeset
|
706 |
|
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
707 |
\begin{proof} |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
708 |
To define the chain maps between the two complexes we will use the following lemma: |
939a4a5b1d80
increased line width; start to write commutative algebra results
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45
diff
changeset
|
709 |
|
939a4a5b1d80
increased line width; start to write commutative algebra results
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diff
changeset
|
710 |
\begin{lemma} |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
711 |
Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
712 |
a basis (e.g.\ blob diagrams or singular simplices). |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
713 |
For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
48
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
714 |
such that $R(c')_* \sub R(c)_*$ whenever $c'$ is a basis element which is part of $\bd c$. |
47
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
715 |
Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
716 |
$f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
939a4a5b1d80
increased line width; start to write commutative algebra results
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45
diff
changeset
|
717 |
\end{lemma} |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
718 |
|
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
719 |
\begin{proof} |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
720 |
\nn{easy, but should probably write the details eventually} |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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45
diff
changeset
|
721 |
\end{proof} |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
45
diff
changeset
|
722 |
|
48
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
723 |
Our first task: For each blob diagram $b$ define a subcomplex $R(b)_* \sub C_*(\Sigma^\infty(M))$ |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
724 |
satisfying the conditions of the above lemma. |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
725 |
If $b$ is a 0-blob diagram, then it is just a $k[t]$ field on $M$, which is a |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
726 |
finite unordered collection of points of $M$ with multiplicities, which is |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
727 |
a point in $\Sigma^\infty(M)$. |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
728 |
Define $R(b)_*$ to be the singular chain complex of this point. |
b7ade62bea27
more commutative algebra stuff
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47
diff
changeset
|
729 |
If $(B, u, r)$ is an $i$-blob diagram, let $D\sub M$ be its support (the union of the blobs). |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
730 |
The path components of $\Sigma^\infty(D)$ are contractible, and these components are indexed |
b7ade62bea27
more commutative algebra stuff
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47
diff
changeset
|
731 |
by the numbers of points in each component of $D$. |
b7ade62bea27
more commutative algebra stuff
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
47
diff
changeset
|
732 |
We may assume that the blob labels $u$ have homogeneous $t$ degree in $k[t]$, and so |
b7ade62bea27
more commutative algebra stuff
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parents:
47
diff
changeset
|
733 |
$u$ picks out a component $X \sub \Sigma^\infty(D)$. |
b7ade62bea27
more commutative algebra stuff
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47
diff
changeset
|
734 |
The field $r$ on $M\setminus D$ can be thought of as a point in $\Sigma^\infty(M\setminus D)$, |
b7ade62bea27
more commutative algebra stuff
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|
735 |
and using this point we can embed $X$ in $\Sigma^\infty(M)$. |
b7ade62bea27
more commutative algebra stuff
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|
736 |
Define $R(B, u, r)_*$ to be the singular chain complex of $X$, thought of as a |
b7ade62bea27
more commutative algebra stuff
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47
diff
changeset
|
737 |
subspace of $\Sigma^\infty(M)$. |
b7ade62bea27
more commutative algebra stuff
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47
diff
changeset
|
738 |
It is easy to see that $R(\cdot)_*$ satisfies the condition on boundaries from the above lemma. |
b7ade62bea27
more commutative algebra stuff
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|
739 |
Thus we have defined (up to homotopy) a map from |
b7ade62bea27
more commutative algebra stuff
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changeset
|
740 |
$\bc_*(M^n, k[t])$ to $C_*(\Sigma^\infty(M))$. |
47
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|
741 |
|
48
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more commutative algebra stuff
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changeset
|
742 |
Next we define, for each simplex $c$ of $C_*(\Sigma^\infty(M))$, a contractible subspace |
b7ade62bea27
more commutative algebra stuff
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changeset
|
743 |
$R(c)_* \sub \bc_*(M^n, k[t])$. |
b7ade62bea27
more commutative algebra stuff
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|
744 |
If $c$ is a 0-simplex we use the identification of the fields $\cC(M)$ and |
b7ade62bea27
more commutative algebra stuff
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changeset
|
745 |
$\Sigma^\infty(M)$ described above. |
b7ade62bea27
more commutative algebra stuff
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|
746 |
Now let $c$ be an $i$-simplex of $\Sigma^j(M)$. |
b7ade62bea27
more commutative algebra stuff
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|
747 |
Choose a metric on $M$, which induces a metric on $\Sigma^j(M)$. |
b7ade62bea27
more commutative algebra stuff
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|
748 |
We may assume that the diameter of $c$ is small --- that is, $C_*(\Sigma^j(M))$ |
b7ade62bea27
more commutative algebra stuff
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|
749 |
is homotopy equivalent to the subcomplex of small simplices. |
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more commutative algebra stuff
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|
750 |
How small? $(2r)/3j$, where $r$ is the radius of injectivity of the metric. |
b7ade62bea27
more commutative algebra stuff
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changeset
|
751 |
Let $T\sub M$ be the ``track" of $c$ in $M$. |
b7ade62bea27
more commutative algebra stuff
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changeset
|
752 |
\nn{do we need to define this precisely?} |
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more commutative algebra stuff
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|
753 |
Choose a neighborhood $D$ of $T$ which is a disjoint union of balls of small diameter. |
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more commutative algebra stuff
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|
754 |
\nn{need to say more precisely how small} |
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|
755 |
Define $R(c)_*$ to be $\bc_*(D, k[t]) \sub \bc_*(M^n, k[t])$. |
b7ade62bea27
more commutative algebra stuff
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|
756 |
This is contractible by \ref{bcontract}. |
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more commutative algebra stuff
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changeset
|
757 |
We can arrange that the boundary/inclusion condition is satisfied if we start with |
b7ade62bea27
more commutative algebra stuff
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changeset
|
758 |
low-dimensional simplices and work our way up. |
b7ade62bea27
more commutative algebra stuff
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changeset
|
759 |
\nn{need to be more precise} |
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more commutative algebra stuff
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changeset
|
760 |
|
b7ade62bea27
more commutative algebra stuff
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changeset
|
761 |
\nn{still to do: show indep of choice of metric; show compositions are homotopic to the identity |
b7ade62bea27
more commutative algebra stuff
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|
762 |
(for this, might need a lemma that says we can assume that blob diameters are small)} |
47
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|
763 |
\end{proof} |
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|
764 |
|
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|
765 |
|
50 | 766 |
\begin{prop} \label{ktcdprop} |
49 | 767 |
The above maps are compatible with the evaluation map actions of $C_*(\Diff(M))$. |
768 |
\end{prop} |
|
769 |
||
770 |
\begin{proof} |
|
771 |
The actions agree in degree 0, and both are compatible with gluing. |
|
772 |
(cf. uniqueness statement in \ref{CDprop}.) |
|
773 |
\end{proof} |
|
774 |
||
775 |
\medskip |
|
776 |
||
777 |
In view of \ref{hochthm}, we have proved that $HH_*(k[t]) \cong C_*(\Sigma^\infty(S^1), k)$, |
|
778 |
and that the cyclic homology of $k[t]$ is related to the action of rotations |
|
779 |
on $C_*(\Sigma^\infty(S^1), k)$. |
|
780 |
\nn{probably should put a more precise statement about cyclic homology and $S^1$ actions in the Hochschild section} |
|
781 |
Let us check this directly. |
|
782 |
||
50 | 783 |
According to \nn{Loday, 3.2.2}, $HH_i(k[t]) \cong k[t]$ for $i=0,1$ and is zero for $i\ge 2$. |
784 |
\nn{say something about $t$-degree? is this in [Loday]?} |
|
785 |
||
49 | 786 |
We can define a flow on $\Sigma^j(S^1)$ by having the points repel each other. |
787 |
The fixed points of this flow are the equally spaced configurations. |
|
50 | 788 |
This defines a map from $\Sigma^j(S^1)$ to $S^1/j$ ($S^1$ modulo a $2\pi/j$ rotation). |
49 | 789 |
The fiber of this map is $\Delta^{j-1}$, the $(j-1)$-simplex, |
790 |
and the holonomy of the $\Delta^{j-1}$ bundle |
|
50 | 791 |
over $S^1/j$ is induced by the cyclic permutation of its $j$ vertices. |
792 |
||
793 |
In particular, $\Sigma^j(S^1)$ is homotopy equivalent to a circle for $j>0$, and |
|
794 |
of course $\Sigma^0(S^1)$ is a point. |
|
795 |
Thus the singular homology $H_i(\Sigma^\infty(S^1))$ has infinitely many generators for $i=0,1$ |
|
796 |
and is zero for $i\ge 2$. |
|
797 |
\nn{say something about $t$-degrees also matching up?} |
|
798 |
||
799 |
By xxxx and \ref{ktcdprop}, |
|
800 |
the cyclic homology of $k[t]$ is the $S^1$-equivariant homology of $\Sigma^\infty(S^1)$. |
|
801 |
Up to homotopy, $S^1$ acts by $j$-fold rotation on $\Sigma^j(S^1) \simeq S^1/j$. |
|
51
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|
802 |
If $k = \z$, $\Sigma^j(S^1)$ contributes the homology of an infinite lens space: $\z$ in degree |
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|
803 |
0, $\z/j \z$ in odd degrees, and 0 in positive even degrees. |
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|
804 |
The point $\Sigma^0(S^1)$ contributes the homology of $BS^1$ which is $\z$ in even |
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|
805 |
degrees and 0 in odd degrees. |
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|
806 |
This agrees with the calculation in \nn{Loday, 3.1.7}. |
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|
807 |
|
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|
808 |
\medskip |
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|
809 |
|
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|
810 |
Next we consider the case $C = k[t_1, \ldots, t_m]$, commutative polynomials in $m$ variables. |
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|
811 |
Let $\Sigma_m^\infty(M)$ be the $m$-colored infinite symmetric power of $M$, that is, configurations |
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|
812 |
of points on $M$ which can have any of $m$ distinct colors but are otherwise indistinguishable. |
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|
813 |
The components of $\Sigma_m^\infty(M)$ are indexed by $m$-tuples of natural numbers |
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|
814 |
corresponding to the number of points of each color of a configuration. |
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|
815 |
A proof similar to that of \ref{sympowerprop} shows that |
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|
816 |
|
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|
817 |
\begin{prop} |
52 | 818 |
$\bc_*(M, k[t_1, \ldots, t_m])$ is homotopy equivalent to $C_*(\Sigma_m^\infty(M), k)$. |
51
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|
819 |
\end{prop} |
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|
820 |
|
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|
821 |
According to \nn{Loday, 3.2.2}, |
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|
822 |
\[ |
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|
823 |
HH_n(k[t_1, \ldots, t_m]) \cong \Lambda^n(k^m) \otimes k[t_1, \ldots, t_m] . |
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|
824 |
\] |
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|
825 |
Let us check that this is also the singular homology of $\Sigma_m^\infty(S^1)$. |
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|
826 |
We will content ourselves with the case $k = \z$. |
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|
827 |
One can define a flow on $\Sigma_m^\infty(S^1)$ where points of the same color repel each other and points of different colors do not interact. |
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|
828 |
This shows that a component $X$ of $\Sigma_m^\infty(S^1)$ is homotopy equivalent |
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|
829 |
to the torus $(S^1)^l$, where $l$ is the number of non-zero entries in the $m$-tuple |
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|
830 |
corresponding to $X$. |
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|
831 |
The homology calculation we desire follows easily from this. |
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|
832 |
|
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|
833 |
\nn{say something about cyclic homology in this case? probably not necessary.} |
49 | 834 |
|
52 | 835 |
\medskip |
49 | 836 |
|
52 | 837 |
Next we consider the case $C = k[t]/t^l$ --- polynomials in $t$ with $t^l = 0$. |
838 |
Define $\Delta_l \sub \Sigma^\infty(M)$ to be those configurations of points with $l$ or |
|
839 |
more points coinciding. |
|
49 | 840 |
|
52 | 841 |
\begin{prop} |
842 |
$\bc_*(M, k[t]/t^l)$ is homotopy equivalent to $C_*(\Sigma^\infty(M), \Delta_l, k)$ |
|
843 |
(relative singular chains with coefficients in $k$). |
|
844 |
\end{prop} |
|
49 | 845 |
|
52 | 846 |
\begin{proof} |
847 |
\nn{...} |
|
848 |
\end{proof} |
|
49 | 849 |
|
850 |
\nn{...} |
|
851 |
||
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|
852 |
|
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853 |
|
22
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|
854 |
|
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|
855 |
\appendix |
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|
856 |
|
98 | 857 |
\input{text/famodiff.tex} |
0 | 858 |
|
75 | 859 |
\section{Comparing definitions of $A_\infty$ algebras} |
76 | 860 |
\label{sec:comparing-A-infty} |
75 | 861 |
In this section, we make contact between the usual definition of an $A_\infty$ algebra and our definition of a topological $A_\infty$ algebra, from Definition \ref{defn:topological-Ainfty-category}. |
862 |
||
863 |
We begin be restricting the data of a topological $A_\infty$ algebra to the standard interval $[0,1]$, which we can alternatively characterise as: |
|
864 |
\begin{defn} |
|
865 |
A \emph{topological $A_\infty$ category on $[0,1]$} $\cC$ has a set of objects $\Obj(\cC)$, and for each $a,b \in \Obj(\cC)$, a chain complex $\cC_{a,b}$, along with |
|
866 |
\begin{itemize} |
|
867 |
\item an action of the operad of $\Obj(\cC)$-labeled cell decompositions |
|
868 |
\item and a compatible action of $\CD{[0,1]}$. |
|
869 |
\end{itemize} |
|
870 |
\end{defn} |
|
871 |
Here the operad of cell decompositions of $[0,1]$ has operations indexed by a finite set of points $0 < x_1< \cdots < x_k < 1$, cutting $[0,1]$ into subintervals. An $X$-labeled cell decomposition labels $\{0, x_1, \ldots, x_k, 1\}$ by $X$. Given two cell decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$, we can compose them to form a new cell decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$ by inserting the points of $\cJ^{(2)}$ linearly into the $m$-th interval of $\cJ^{(1)}$. In the $X$-labeled case, we insist that the appropriate labels match up. Saying we have an action of this operad means that for each labeled cell decomposition $0 < x_1< \cdots < x_k < 1$, $a_0, \ldots, a_{k+1} \subset \Obj(\cC)$, there is a chain map $$\cC_{a_0,a_1} \tensor \cdots \tensor \cC_{a_k,a_{k+1}} \to \cC(a_0,a_{k+1})$$ and these chain maps compose exactly as the cell decompositions. |
|
872 |
An action of $\CD{[0,1]}$ is compatible with an action of the cell decomposition operad if given a decomposition $\pi$, and a family of diffeomorphisms $f \in \CD{[0,1]}$ which is supported on the subintervals determined by $\pi$, then the two possible operations (glue intervals together, then apply the diffeomorphisms, or apply the diffeormorphisms separately to the subintervals, then glue) commute (as usual, up to a weakly unique homotopy). |
|
873 |
||
874 |
Translating between these definitions is straightforward. To restrict to the standard interval, define $\cC_{a,b} = \cC([0,1];a,b)$. Given a cell decomposition $0 < x_1< \cdots < x_k < 1$, we use the map (suppressing labels) |
|
875 |
$$\cC([0,1])^{\tensor k+1} \to \cC([0,x_1]) \tensor \cdots \tensor \cC[x_k,1] \to \cC([0,1])$$ |
|
876 |
where the factors of the first map are induced by the linear isometries $[0,1] \to [x_i, x_{i+1}]$, and the second map is just gluing. The action of $\CD{[0,1]}$ carries across, and is automatically compatible. Going the other way, we just declare $\cC(J;a,b) = \cC_{a,b}$, pick a diffeomorphism $\phi_J : J \isoto [0,1]$ for every interval $J$, define the gluing map $\cC(J_1) \tensor \cC(J_2) \to \cC(J_1 \cup J_2)$ by the first applying the cell decomposition map for $0 < \frac{1}{2} < 1$, then the self-diffeomorphism of $[0,1]$ given by $\frac{1}{2} (\phi_{J_1} \cup (1+ \phi_{J_2})) \circ \phi_{J_1 \cup J_2}^{-1}$. You can readily check that this gluing map is associative on the nose. \todo{really?} |
|
877 |
||
878 |
%First recall the \emph{coloured little intervals operad}. Given a set of labels $\cL$, the operations are indexed by \emph{decompositions of the interval}, each of which is a collection of disjoint subintervals $\{(a_i,b_i)\}_{i=1}^k$ of $[0,1]$, along with a labeling of the complementary regions by $\cL$, $\{l_0, \ldots, l_k\}$. Given two decompositions $\cJ^{(1)}$ and $\cJ^{(2)}$, and an index $m$ such that $l^{(1)}_{m-1} = l^{(2)}_0$ and $l^{(1)}_{m} = l^{(2)}_{k^{(2)}}$, we can form a new decomposition by inserting the intervals of $\cJ^{(2)}$ linearly inside the $m$-th interval of $\cJ^{(1)}$. We call the resulting decomposition $\cJ^{(1)} \circ_m \cJ^{(2)}$. |
|
879 |
||
880 |
%\begin{defn} |
|
881 |
%A \emph{topological $A_\infty$ category} $\cC$ has a set of objects $\Obj(\cC)$ and for each $a,b \in \Obj(\cC)$ a chain complex $\cC_{a,b}$, along with a compatible `composition map' and an `action of families of diffeomorphisms'. |
|
882 |
||
883 |
%A \emph{composition map} $f$ is a family of chain maps, one for each decomposition of the interval, $f_\cJ : A^{\tensor k} \to A$, making $\cC$ into a category over the coloured little intervals operad, with labels $\cL = \Obj(\cC)$. Thus the chain maps satisfy the identity |
|
884 |
%\begin{equation*} |
|
885 |
%f_{\cJ^{(1)} \circ_m \cJ^{(2)}} = f_{\cJ^{(1)}} \circ (\id^{\tensor m-1} \tensor f_{\cJ^{(2)}} \tensor \id^{\tensor k^{(1)} - m}). |
|
886 |
%\end{equation*} |
|
887 |
||
888 |
%An \emph{action of families of diffeomorphisms} is a chain map $ev: \CD{[0,1]} \tensor A \to A$, such that |
|
889 |
%\begin{enumerate} |
|
890 |
%\item The diagram |
|
891 |
%\begin{equation*} |
|
892 |
%\xymatrix{ |
|
893 |
%\CD{[0,1]} \tensor \CD{[0,1]} \tensor A \ar[r]^{\id \tensor ev} \ar[d]^{\circ \tensor \id} & \CD{[0,1]} \tensor A \ar[d]^{ev} \\ |
|
894 |
%\CD{[0,1]} \tensor A \ar[r]^{ev} & A |
|
895 |
%} |
|
896 |
%\end{equation*} |
|
897 |
%commutes up to weakly unique homotopy. |
|
898 |
%\item If $\phi \in \Diff([0,1])$ and $\cJ$ is a decomposition of the interval, we obtain a new decomposition $\phi(\cJ)$ and a collection $\phi_m \in \Diff([0,1])$ of diffeomorphisms obtained by taking the restrictions $\restrict{\phi}{[a_m,b_m]} : [a_m,b_m] \to [\phi(a_m),\phi(b_m)]$ and pre- and post-composing these with the linear diffeomorphisms $[0,1] \to [a_m,b_m]$ and $[\phi(a_m),\phi(b_m)] \to [0,1]$. We require that |
|
899 |
%\begin{equation*} |
|
900 |
%\phi(f_\cJ(a_1, \cdots, a_k)) = f_{\phi(\cJ)}(\phi_1(a_1), \cdots, \phi_k(a_k)). |
|
901 |
%\end{equation*} |
|
902 |
%\end{enumerate} |
|
903 |
%\end{defn} |
|
904 |
||
905 |
From a topological $A_\infty$ category on $[0,1]$ $\cC$ we can produce a `conventional' $A_\infty$ category $(A, \{m_k\})$ as defined in, for example, \cite{MR1854636}. We'll just describe the algebra case (that is, a category with only one object), as the modifications required to deal with multiple objects are trivial. Define $A = \cC$ as a chain complex (so $m_1 = d$). Define $m_2 : A\tensor A \to A$ by $f_{\{(0,\frac{1}{2}),(\frac{1}{2},1)\}}$. To define $m_3$, we begin by taking the one parameter family $\phi_3$ of diffeomorphisms of $[0,1]$ that interpolates linearly between the identity and the piecewise linear diffeomorphism taking $\frac{1}{4}$ to $\frac{1}{2}$ and $\frac{1}{2}$ to $\frac{3}{4}$, and then define |
|
906 |
\begin{equation*} |
|
907 |
m_3(a,b,c) = ev(\phi_3, m_2(m_2(a,b), c)). |
|
908 |
\end{equation*} |
|
909 |
||
910 |
It's then easy to calculate that |
|
911 |
\begin{align*} |
|
912 |
d(m_3(a,b,c)) & = ev(d \phi_3, m_2(m_2(a,b),c)) - ev(\phi_3 d m_2(m_2(a,b), c)) \\ |
|
913 |
& = ev( \phi_3(1), m_2(m_2(a,b),c)) - ev(\phi_3(0), m_2 (m_2(a,b),c)) - \\ & \qquad - ev(\phi_3, m_2(m_2(da, b), c) + (-1)^{\deg a} m_2(m_2(a, db), c) + \\ & \qquad \quad + (-1)^{\deg a+\deg b} m_2(m_2(a, b), dc) \\ |
|
914 |
& = m_2(a , m_2(b,c)) - m_2(m_2(a,b),c) - \\ & \qquad - m_3(da,b,c) + (-1)^{\deg a + 1} m_3(a,db,c) + \\ & \qquad \quad + (-1)^{\deg a + \deg b + 1} m_3(a,b,dc), \\ |
|
915 |
\intertext{and thus that} |
|
916 |
m_1 \circ m_3 & = m_2 \circ (\id \tensor m_2) - m_2 \circ (m_2 \tensor \id) - \\ & \qquad - m_3 \circ (m_1 \tensor \id \tensor \id) - m_3 \circ (\id \tensor m_1 \tensor \id) - m_3 \circ (\id \tensor \id \tensor m_1) |
|
917 |
\end{align*} |
|
918 |
as required (c.f. \cite[p. 6]{MR1854636}). |
|
919 |
\todo{then the general case.} |
|
920 |
We won't describe a reverse construction (producing a topological $A_\infty$ category from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. |
|
921 |
||
76 | 922 |
\section{Morphisms and duals of topological $A_\infty$ modules} |
923 |
\label{sec:A-infty-hom-and-duals}% |
|
924 |
||
925 |
\begin{defn} |
|
926 |
If $\cM$ and $\cN$ are topological $A_\infty$ left modules over a topological $A_\infty$ category $\cC$, then a morphism $f: \cM \to \cN$ consists of a chain map $f:\cM(J,p;b) \to \cN(J,p;b)$ for each right marked interval $(J,p)$ with a boundary condition $b$, such that for each interval $J'$ the diagram |
|
927 |
\begin{equation*} |
|
928 |
\xymatrix{ |
|
929 |
\cC(J';a,b) \tensor \cM(J,p;b) \ar[r]^{\text{gl}} \ar[d]^{\id \tensor f} & \cM(J' cup J,a) \ar[d]^f \\ |
|
930 |
\cC(J';a,b) \tensor \cN(J,p;b) \ar[r]^{\text{gl}} & \cN(J' cup J,a) |
|
931 |
} |
|
932 |
\end{equation*} |
|
933 |
commutes on the nose, and the diagram |
|
934 |
\begin{equation*} |
|
935 |
\xymatrix{ |
|
936 |
\CD{(J,p) \to (J',p')} \tensor \cM(J,p;a) \ar[r]^{\text{ev}} \ar[d]^{\id \tensor f} & \cM(J',p';a) \ar[d]^f \\ |
|
937 |
\CD{(J,p) \to (J',p')} \tensor \cN(J,p;a) \ar[r]^{\text{ev}} & \cN(J',p';a) \\ |
|
938 |
} |
|
939 |
\end{equation*} |
|
940 |
commutes up to a weakly unique homotopy. |
|
941 |
\end{defn} |
|
942 |
||
943 |
The variations required for right modules and bimodules should be obvious. |
|
944 |
||
945 |
\todo{duals} |
|
946 |
\todo{ the functors $\hom_{\lmod{\cC}}\left(\cM \to -\right)$ and $\cM^* \tensor_{\cC} -$ from $\lmod{\cC}$ to $\Vect$ are naturally isomorphic} |
|
75 | 947 |
|
948 |
||
55 | 949 |
\input{text/obsolete.tex} |
47
939a4a5b1d80
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|
950 |
|
22
ada83e7228eb
rearranging; stating all the "properties" up front
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|
951 |
% ---------------------------------------------------------------- |
23
7b0a43bdd3c4
writing definitions of topological a_\infty categories, modules, etc.
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|
952 |
%\newcommand{\urlprefix}{} |
7b0a43bdd3c4
writing definitions of topological a_\infty categories, modules, etc.
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|
953 |
\bibliographystyle{plain} |
22
ada83e7228eb
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|
954 |
%Included for winedt: |
ada83e7228eb
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|
955 |
%input "bibliography/bibliography.bib" |
ada83e7228eb
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|
956 |
\bibliography{bibliography/bibliography} |
ada83e7228eb
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|
957 |
% ---------------------------------------------------------------- |
7
4ef2f77a4652
small to medium sized changes
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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changeset
|
958 |
|
22
ada83e7228eb
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|
959 |
This paper is available online at \arxiv{?????}, and at |
47
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
960 |
\url{http://tqft.net/blobs}, |
939a4a5b1d80
increased line width; start to write commutative algebra results
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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|
961 |
and at \url{http://canyon23.net/math/}. |
7
4ef2f77a4652
small to medium sized changes
kevin@6e1638ff-ae45-0410-89bd-df963105f760
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changeset
|
962 |
|
22
ada83e7228eb
rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
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|
963 |
% A GTART necessity: |
ada83e7228eb
rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
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|
964 |
% \Addresses |
ada83e7228eb
rearranging; stating all the "properties" up front
scott@6e1638ff-ae45-0410-89bd-df963105f760
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|
965 |
% ---------------------------------------------------------------- |
ada83e7228eb
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scott@6e1638ff-ae45-0410-89bd-df963105f760
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|
966 |
\end{document} |
ada83e7228eb
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|
967 |
% ---------------------------------------------------------------- |
0 | 968 |
|
969 |
||
970 |
||
971 |
||
972 |
%Recall that for $n$-category picture fields there is an evaluation map |
|
973 |
%$m: \bc_0(B^n; c, c') \to \mor(c, c')$. |
|
974 |
%If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain |
|
975 |
%map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. |