# HG changeset patch # User scott@6e1638ff-ae45-0410-89bd-df963105f760 # Date 1209005794 0 # Node ID 15e6335ff1d4a6c44d3c7c60d3b704c48c86fad7 # Parent 4ef2f77a4652a8781f6f5007155bcb264016bcf4 ... diff -r 4ef2f77a4652 -r 15e6335ff1d4 00README.XXX --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/00README.XXX Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +nohypertex diff -r 4ef2f77a4652 -r 15e6335ff1d4 ant-arxiv.bat --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ant-arxiv.bat Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +ant arxiv diff -r 4ef2f77a4652 -r 15e6335ff1d4 ant-arxiv.sh --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ant-arxiv.sh Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +ant arxiv diff -r 4ef2f77a4652 -r 15e6335ff1d4 ant-eps-diagrams.bat --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ant-eps-diagrams.bat Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +ant eps-diagrams diff -r 4ef2f77a4652 -r 15e6335ff1d4 ant-pdf.bat --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ant-pdf.bat Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +ant pdf diff -r 4ef2f77a4652 -r 15e6335ff1d4 ant-pdf.sh --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ant-pdf.sh Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +ant pdf diff -r 4ef2f77a4652 -r 15e6335ff1d4 bibliography/bibliography.bib --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/bibliography/bibliography.bib Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,405 @@ +@PREAMBLE{ "\newcommand{\noopsort}[1]{}" } + +@STRING{CUP = {Cambridge University Press}} +@STRING{OUP = {Oxford University Press}} +@STRING{PUP = {Princeton University Press}} +@STRING{SV = {Springer-Verlag}} +@STRING{AP = {Academic Press}} +@STRING{AMS = {American Mathematical Society}} + +@article {MR1917056, + AUTHOR = {Bar-Natan, Dror}, + TITLE = {On {K}hovanov's categorification of the {J}ones polynomial}, + JOURNAL = {Algebr. Geom. Topol.}, + FJOURNAL = {Algebraic \& Geometric Topology}, + VOLUME = {2}, + YEAR = {2002}, + PAGES = {337--370 (electronic)}, + ISSN = {1472-2747}, + MRCLASS = {57M27}, + MRNUMBER = {MR1917056 (2003h:57014)}, +MRREVIEWER = {Jacob Andrew Rasmussen}, + note = {arXiv:\arxiv{math.QA/0201043}}, +} + +@incollection {MR2147420, + AUTHOR = {Bar-Natan, Dror}, + TITLE = {Khovanov homology for knots and links with up to 11 crossings}, + BOOKTITLE = {Advances in topological quantum field theory}, + SERIES = {NATO Sci. Ser. II Math. Phys. Chem.}, + VOLUME = {179}, + PAGES = {167--241}, + PUBLISHER = {Kluwer Acad. Publ.}, + ADDRESS = {Dordrecht}, + YEAR = {2004}, + MRCLASS = {57M27}, + MRNUMBER = {MR2147420 (2006c:57009)}, +MRREVIEWER = {Marta M. Asaeda}, +} + +@article {MR2174270, + AUTHOR = {Bar-Natan, Dror}, + TITLE = {Khovanov's homology for tangles and cobordisms}, + JOURNAL = {Geom. Topol.}, + FJOURNAL = {Geometry and Topology}, + VOLUME = {9}, + YEAR = {2005}, + PAGES = {1443--1499 (electronic)}, + ISSN = {1465-3060}, + MRCLASS = {57M27 (57M25)}, + MRNUMBER = {MR2174270}, + note = {arXiv:\arxiv{math.GT/0410495}}, +} + +@article {MR1680395, + AUTHOR = {Khovanov, Mikhail and Kuperberg, Greg}, + TITLE = {Web bases for {${\rm sl}(3)$} are not dual canonical}, + JOURNAL = {Pacific J. Math.}, + FJOURNAL = {Pacific Journal of Mathematics}, + VOLUME = {188}, + YEAR = {1999}, + NUMBER = {1}, + PAGES = {129--153}, + ISSN = {0030-8730}, + CODEN = {PJMAAI}, + MRCLASS = {17B37 (22E60 57M27 81R05)}, + MRNUMBER = {MR1680395 (2000j:17023a)}, +MRREVIEWER = {Robert J. Marsh}, + note = {arXiv:\arxiv{q-alg/9712046}}, +} + +@article {MR1740682, + AUTHOR = {Khovanov, Mikhail}, + TITLE = {A categorification of the {J}ones polynomial}, + JOURNAL = {Duke Math. J.}, + FJOURNAL = {Duke Mathematical Journal}, + VOLUME = {101}, + YEAR = {2000}, + NUMBER = {3}, + PAGES = {359--426}, + ISSN = {0012-7094}, + CODEN = {DUMJAO}, + MRCLASS = {57M27 (57R56)}, + MRNUMBER = {MR1740682 (2002j:57025)}, +} + +@article {MR1928174, + AUTHOR = {Khovanov, Mikhail}, + TITLE = {A functor-valued invariant of tangles}, + JOURNAL = {Algebr. Geom. Topol.}, + FJOURNAL = {Algebraic \& Geometric Topology}, + VOLUME = {2}, + YEAR = {2002}, + PAGES = {665--741 (electronic)}, + ISSN = {1472-2747}, + MRCLASS = {57M27 (57R56)}, + MRNUMBER = {MR1928174 (2004d:57016)}, +MRREVIEWER = {Jacob Andrew Rasmussen}, + note = {arXiv:\arxiv{math.GT/0103190}}, +} + +@article {MR2034399, + AUTHOR = {Khovanov, Mikhail}, + TITLE = {Patterns in knot cohomology. {I}}, + JOURNAL = {Experiment. Math.}, + FJOURNAL = {Experimental Mathematics}, + VOLUME = {12}, + YEAR = {2003}, + NUMBER = {3}, + PAGES = {365--374}, + ISSN = {1058-6458}, + MRCLASS = {57M27 (18G60 57M25 57R56)}, + MRNUMBER = {MR2034399 (2004m:57022)}, +MRREVIEWER = {Jacob Andrew Rasmussen}, +} + +@article {MR2100691, + AUTHOR = {Khovanov, Mikhail}, + TITLE = {sl(3) link homology}, + JOURNAL = {Algebr. Geom. Topol.}, + FJOURNAL = {Algebraic \& Geometric Topology}, + VOLUME = {4}, + YEAR = {2004}, + PAGES = {1045--1081 (electronic)}, + ISSN = {1472-2747}, + MRCLASS = {57M27 (18G60 57R56)}, + MRNUMBER = {MR2100691 (2005g:57032)}, +MRREVIEWER = {Justin Sawon}, + note = {arXiv:\arxiv{math.QA/0304375}}, +} + +@article {MR2124557, + AUTHOR = {Khovanov, Mikhail}, + TITLE = {Categorifications of the colored {J}ones polynomial}, + JOURNAL = {J. Knot Theory Ramifications}, + FJOURNAL = {Journal of Knot Theory and its Ramifications}, + VOLUME = {14}, + YEAR = {2005}, + NUMBER = {1}, + PAGES = {111--130}, + ISSN = {0218-2165}, + MRCLASS = {57M27}, + MRNUMBER = {MR2124557 (2006a:57016)}, +MRREVIEWER = {Marta M. Asaeda}, +} + +@article {MR2171235, + AUTHOR = {Khovanov, Mikhail}, + TITLE = {An invariant of tangle cobordisms}, + JOURNAL = {Trans. Amer. Math. Soc.}, + FJOURNAL = {Transactions of the American Mathematical Society}, + VOLUME = {358}, + YEAR = {2006}, + NUMBER = {1}, + PAGES = {315--327 (electronic)}, + ISSN = {0002-9947}, + CODEN = {TAMTAM}, + MRCLASS = {57Q45}, + MRNUMBER = {MR2171235}, + note = {arXiv:\arxiv{math.GT/0207264}}, +} + +@misc{math.QA/0401268, + title = {{Matrix factorizations and link homology}}, + author = {Mikhail Khovanov and Lev Rozansky}, + note = {arXiv:\arxiv{math.QA/0401268}}} + +@misc{math.QA/0505056, + title = {{Matrix factorizations and link homology II}}, + author = {Mikhail Khovanov and Lev Rozansky}, + note = {arXiv:\arxiv{math.QA/0505056}}} + +@incollection {MR2048108, + AUTHOR = {Thurston, Dylan P.}, + TITLE = {The algebra of knotted trivalent graphs and {T}uraev's shadow + world}, + BOOKTITLE = {Invariants of knots and 3-manifolds (Kyoto, 2001)}, + SERIES = {Geom. Topol. Monogr.}, + VOLUME = {4}, + PAGES = {337--362 (electronic)}, + PUBLISHER = {Geom. Topol. Publ., Coventry}, + YEAR = {2002}, + MRCLASS = {57M25 (57M15 57M20 57Q40)}, + MRNUMBER = {MR2048108 (2005c:57010)}, +MRREVIEWER = {Marko Kranjc}, + note = {arXiv:\arxiv{math.GT/0311458}}, +} + +@article {MR1403861, + AUTHOR = {Kuperberg, Greg}, + TITLE = {Spiders for rank {$2$} {L}ie algebras}, + JOURNAL = {Comm. Math. Phys.}, + FJOURNAL = {Communications in Mathematical Physics}, + VOLUME = {180}, + YEAR = {1996}, + NUMBER = {1}, + PAGES = {109--151}, + ISSN = {0010-3616}, + CODEN = {CMPHAY}, + MRCLASS = {17B10 (22E60 81R05)}, + MRNUMBER = {MR1403861 (97f:17005)}, +MRREVIEWER = {Stefano Capparelli}, + note = {arXiv:\arxiv{q-alg/9712003}}, +} + +@misc{math.GT/0606318, + title = {{Fast Khovanov Homology Computations}}, + author = {Dror Bar-Natan}, + note = {arXiv:\arxiv{math.GT/0606318}}, + url = {http://www.math.toronto.edu/~drorbn/papers/FastKh/} + } + +@misc{math.GT/0603347, + title = {{On the Algebraic Structure of Bar-Natan's Universal + Geometric Complex and the Geometric Structure of Khovanov + Link Homology Theories}}, + author = {Gad Naot}, + note = {arXiv:\arxiv{math.GT/0603347}}} + +@misc{math.QA/9909027, + title = {{Planar algebras, I}}, + author = {Vaughan F. R. Jones}, + note = {arXiv:\arxiv{math.QA/9909027}}} + +@article {MR1217386, + AUTHOR = {Turaev, V. and Wenzl, H.}, + TITLE = {Quantum invariants of {$3$}-manifolds associated with + classical simple {L}ie algebras}, + JOURNAL = {Internat. J. Math.}, + FJOURNAL = {International Journal of Mathematics}, + VOLUME = {4}, + YEAR = {1993}, + NUMBER = {2}, + PAGES = {323--358}, + ISSN = {0129-167X}, + MRCLASS = {57M25 (17B37 57N10)}, + MRNUMBER = {MR1217386 (94i:57019)}, +MRREVIEWER = {Toshitake Kohno}, +} + +@article {MR1470857, + AUTHOR = {Wenzl, Hans}, + TITLE = {{$C\sp *$} tensor categories from quantum groups}, + JOURNAL = {J. Amer. Math. Soc.}, + FJOURNAL = {Journal of the American Mathematical Society}, + VOLUME = {11}, + YEAR = {1998}, + NUMBER = {2}, + PAGES = {261--282}, + ISSN = {0894-0347}, + MRCLASS = {46L89 (17B37 46L05)}, + MRNUMBER = {MR1470857 (98k:46123)}, +MRREVIEWER = {Andrzej Sitarz}, + eprint = {\url{http://ftp.ams.org/jams/1998-11-02/S0894-0347-98-00253-7/S0894-0347-98-00253-7.pdf}}, +} + +@misc{math.GT/0603307, + title = {{The universal sl3-link homology}}, + author = {Marco Mackaay and Pedro Vaz}, + note = {arXiv:\arxiv{math.GT/0603307}}} + + @misc{wiki:Grothendieck-group, + author = "Wikipedia", + title = "Grothendieck group --- Wikipedia{,} The Free Encyclopedia", + year = "2006", + note = "[\href{http://en.wikipedia.org/w/index.php?title=Grothendieck_group&oldid=52451663}{Online}; accessed 30-June-2006]" + } + + @misc{wiki:Invariant-basis-number, + author = "Wikipedia", + title = "Invariant basis number --- Wikipedia{,} The Free Encyclopedia", + year = "2006", + note = "[\href{http://en.wikipedia.org/w/index.php?title=Invariant_basis_number&oldid=85220456}{Online}; accessed 24-December-2006]" + } + +@article {MR1403351, + AUTHOR = {Etingof, Pavel and Kazhdan, David}, + TITLE = {Quantization of {L}ie bialgebras. {I}}, + JOURNAL = {Selecta Math. (N.S.)}, + FJOURNAL = {Selecta Mathematica. New Series}, + VOLUME = {2}, + YEAR = {1996}, + NUMBER = {1}, + PAGES = {1--41}, + ISSN = {1022-1824}, + CODEN = {SMATF6}, + MRCLASS = {17B37 (16W30 18D20 81R50)}, + MRNUMBER = {MR1403351 (97f:17014)}, +MRREVIEWER = {Yu. N. Bespalov}, +} + +@article {MR1669953, + AUTHOR = {Etingof, Pavel and Kazhdan, David}, + TITLE = {Quantization of {L}ie bialgebras. {II}}, + JOURNAL = {Selecta Math. (N.S.)}, + FJOURNAL = {Selecta Mathematica. New Series}, + VOLUME = {4}, + YEAR = {1998}, + NUMBER = {2}, + PAGES = {213--231, 233--269}, + ISSN = {1022-1824}, + CODEN = {SMATF6}, + MRCLASS = {17B62 (16W35 17B37 18D20)}, + MRNUMBER = {MR1669953 (2000i:17033)}, +MRREVIEWER = {Benjamin David Enriquez}, + note = {arXiv:\arxiv{q-alg/9701038}}, +} + +@article {MR1771217, + AUTHOR = {Etingof, Pavel and Kazhdan, David}, + TITLE = {Quantization of {L}ie bialgebras. {IV}. {T}he coinvariant + construction and the quantum {KZ} equations}, + JOURNAL = {Selecta Math. (N.S.)}, + FJOURNAL = {Selecta Mathematica. New Series}, + VOLUME = {6}, + YEAR = {2000}, + NUMBER = {1}, + PAGES = {79--104}, + ISSN = {1022-1824}, + CODEN = {SMATF6}, + MRCLASS = {17B37 (32G34 81R50)}, + MRNUMBER = {MR1771217 (2002i:17021)}, +} + +@article {MR1771218, + AUTHOR = {Etingof, Pavel and Kazhdan, David}, + TITLE = {Quantization of {L}ie bialgebras. {V}. {Q}uantum vertex + operator algebras}, + JOURNAL = {Selecta Math. (N.S.)}, + FJOURNAL = {Selecta Mathematica. New Series}, + VOLUME = {6}, + YEAR = {2000}, + NUMBER = {1}, + PAGES = {105--130}, + ISSN = {1022-1824}, + CODEN = {SMATF6}, + MRCLASS = {17B37 (17B69 81R50)}, + MRNUMBER = {MR1771218 (2002i:17022)}, +} + +@article {MR2253455, + AUTHOR = {Bar-Natan, Dror and Morrison, Scott}, + TITLE = {The {K}aroubi envelope and {L}ee's degeneration of {K}hovanov + homology}, + JOURNAL = {Algebr. Geom. Topol.}, + FJOURNAL = {Algebraic \& Geometric Topology}, + VOLUME = {6}, + YEAR = {2006}, + PAGES = {1459--1469 (electronic)}, + ISSN = {1472-2747}, + MRCLASS = {57M27 (18E05 57M25)}, + MRNUMBER = {MR2253455}, + note = {arXiv:\arxiv{math.GT/0606542}}, +} + +%S! fix this citation! +@article{morrison-walker, + AUTHOR = {Scott Morrison and Kevin Walker}, + TITLE = {Fixing the functoriality of Khovanov homology}, + note = {\url{http://scott-morrison.org/functoriality}}, +} + +@misc{kw:tqft, + AUTHOR = {Walker, Kevin}, + TITLE = {Topological Quantum Field Theories}, + URL = {http://canyon23.net/math/tc.pdf}, +} + +@article{math.GT/0206303, + title = {{An invariant of link cobordisms from Khovanov homology}}, + author = {Magnus Jacobsson}, + journal = {Algebr. Geom. Topol.}, + volume = 4, + year = 2004, + pages = {1211--1251}, + note = {arXiv:\arxiv{math.GT/0206303}}} + +@misc{math.GT/0610650, + title = {{Khovanov-Rozansky homology via a canopolis formalism}}, + author = {Ben Webster}, + note = {arXiv:\arxiv{math.GT/0610650}}} + +@misc{green-implementation, + title = {{JavaKh}}, + author= {Jeremey Green}, + note = {\url{http://katlas.math.toronto.edu/wiki/Khovanov_Homology}}, +} + +@misc{ortiz-navarro, + title = {{Khovanov Homology and Reidemeister Torsion}}, + author ={Juan Ariel Ortiz-Navarro and Chris Truman}, + note = {a talk at the 2006 Toronto CMS meeting, slides at \url{http://www.math.uiowa.edu/~jortizna/Present-CMS-06.pdf}}, +} + +@book {MR1438306, + AUTHOR = {Gelfand, Sergei I. and Manin, Yuri I.}, + TITLE = {Methods of homological algebra}, + NOTE = {Translated from the 1988 Russian original}, + PUBLISHER = {Springer-Verlag}, + ADDRESS = {Berlin}, + YEAR = {1996}, + PAGES = {xviii+372}, + ISBN = {3-540-54746-0}, + MRCLASS = {18-02 (18Exx 18Gxx 55U35)}, + MRNUMBER = {MR1438306 (97j:18001)}, +} diff -r 4ef2f77a4652 -r 15e6335ff1d4 blob1.pdf Binary file blob1.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 blob1.tex --- a/blob1.tex Tue Apr 22 05:13:02 2008 +0000 +++ b/blob1.tex Thu Apr 24 02:56:34 2008 +0000 @@ -1,14 +1,14 @@ -\documentclass[11pt,leqno]{article} +\documentclass[11pt,leqno]{amsart} -\usepackage{amsmath,amssymb,amsthm} - -\usepackage[all]{xy} +\newcommand{\pathtotrunk}{./} +\input{text/article_preamble.tex} +\input{text/top_matter.tex} % test edit #3 %%%%% excerpts from my include file of standard macros -\def\bc{{\cal B}} +\def\bc{{\mathcal B}} \def\z{\mathbb{Z}} \def\r{\mathbb{R}} @@ -38,23 +38,23 @@ % tricky way to iterate macros over a list \def\semicolon{;} \def\applytolist#1{ - \expandafter\def\csname multi#1\endcsname##1{ - \def\multiack{##1}\ifx\multiack\semicolon - \def\next{\relax} - \else - \csname #1\endcsname{##1} - \def\next{\csname multi#1\endcsname} - \fi - \next} - \csname multi#1\endcsname} + \expandafter\def\csname multi#1\endcsname##1{ + \def\multiack{##1}\ifx\multiack\semicolon + \def\next{\relax} + \else + \csname #1\endcsname{##1} + \def\next{\csname multi#1\endcsname} + \fi + \next} + \csname multi#1\endcsname} % \def\cA{{\cal A}} for A..Z -\def\calc#1{\expandafter\def\csname c#1\endcsname{{\cal #1}}} +\def\calc#1{\expandafter\def\csname c#1\endcsname{{\mathcal #1}}} \applytolist{calc}QWERTYUIOPLKJHGFDSAZXCVBNM; % \DeclareMathOperator{\pr}{pr} etc. \def\declaremathop#1{\expandafter\DeclareMathOperator\csname #1\endcsname{#1}} -\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{End}{Hom}{Mat}{Tet}{cat}{Diff}{sign}; +\applytolist{declaremathop}{pr}{im}{id}{gl}{tr}{rot}{Eq}{obj}{mor}{ob}{Rep}{Tet}{cat}{Diff}{sign}; @@ -74,12 +74,6 @@ \@addtoreset{equation}{section} \gdef\theequation{\thesection.\arabic{equation}} \makeatother -\newtheorem{thm}[equation]{Theorem} -\newtheorem{prop}[equation]{Proposition} -\newtheorem{lemma}[equation]{Lemma} -\newtheorem{cor}[equation]{Corollary} -\newtheorem{defn}[equation]{Definition} - \maketitle @@ -88,10 +82,10 @@ (motivation, summary/outline, etc.) -(motivation: +(motivation: (1) restore exactness in pictures-mod-relations; (1') add relations-amongst-relations etc. to pictures-mod-relations; -(2) want answer independent of handle decomp (i.e. don't +(2) want answer independent of handle decomp (i.e. don't just go from coend to derived coend (e.g. Hochschild homology)); (3) ... ) @@ -102,35 +96,35 @@ Fix a top dimension $n$. -A {\it system of fields} +A {\it system of fields} \nn{maybe should look for better name; but this is the name I use elsewhere} is a collection of functors $\cC$ from manifolds of dimension $n$ or less to sets. These functors must satisfy various properties (see KW TQFT notes for details). -For example: +For example: there is a canonical identification $\cC(X \du Y) = \cC(X) \times \cC(Y)$; there is a restriction map $\cC(X) \to \cC(\bd X)$; gluing manifolds corresponds to fibered products of fields; -given a field $c \in \cC(Y)$ there is a ``product field" +given a field $c \in \cC(Y)$ there is a ``product field" $c\times I \in \cC(Y\times I)$; ... \nn{should eventually include full details of definition of fields.} -\nn{note: probably will suppress from notation the distinction +\nn{note: probably will suppress from notation the distinction between fields and their (orientation-reversal) duals} \nn{remark that if top dimensional fields are not already linear then we will soon linearize them(?)} -The definition of a system of fields is intended to generalize +The definition of a system of fields is intended to generalize the relevant properties of the following two examples of fields. The first example: Fix a target space $B$ and define $\cC(X)$ (where $X$ -is a manifold of dimension $n$ or less) to be the set of +is a manifold of dimension $n$ or less) to be the set of all maps from $X$ to $B$. The second example will take longer to explain. -Given an $n$-category $C$ with the right sort of duality -(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), +Given an $n$-category $C$ with the right sort of duality +(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), we can construct a system of fields as follows. Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ with codimension $i$ cells labeled by $i$-morphisms of $C$. @@ -149,18 +143,18 @@ an object (0-morphism) of the 1-category $C$. A field on a 1-manifold $S$ consists of \begin{itemize} - \item A cell decomposition of $S$ (equivalently, a finite collection + \item A cell decomposition of $S$ (equivalently, a finite collection of points in the interior of $S$); - \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) + \item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) by an object (0-morphism) of $C$; - \item a transverse orientation of each 0-cell, thought of as a choice of + \item a transverse orientation of each 0-cell, thought of as a choice of ``domain" and ``range" for the two adjacent 1-cells; and - \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with + \item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with domain and range determined by the transverse orientation and the labelings of the 1-cells. \end{itemize} If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels -of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the +of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the interior of $S$, each transversely oriented and each labeled by an element (1-morphism) of the algebra. @@ -175,19 +169,19 @@ A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. A field on a 2-manifold $Y$ consists of \begin{itemize} - \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such + \item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such that each component of the complement is homeomorphic to a disk); - \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) + \item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) by a 0-morphism of $C$; - \item a transverse orientation of each 1-cell, thought of as a choice of + \item a transverse orientation of each 1-cell, thought of as a choice of ``domain" and ``range" for the two adjacent 2-cells; - \item a labeling of each 1-cell by a 1-morphism of $C$, with -domain and range determined by the transverse orientation of the 1-cell + \item a labeling of each 1-cell by a 1-morphism of $C$, with +domain and range determined by the transverse orientation of the 1-cell and the labelings of the 2-cells; - \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood + \item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped to $\pm 1 \in S^1$; and - \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range + \item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range determined by the labelings of the 1-cells and the parameterizations of the previous bullet. \end{itemize} @@ -195,10 +189,10 @@ For general $n$, a field on a $k$-manifold $X^k$ consists of \begin{itemize} - \item A cell decomposition of $X$; - \item an explicit general position homeomorphism from the link of each $j$-cell + \item A cell decomposition of $X$; + \item an explicit general position homeomorphism from the link of each $j$-cell to the boundary of the standard $(k-j)$-dimensional bihedron; and - \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with + \item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with domain and range determined by the labelings of the link of $j$-cell. \end{itemize} @@ -208,10 +202,10 @@ \medskip -For top dimensional ($n$-dimensional) manifolds, we're actually interested +For top dimensional ($n$-dimensional) manifolds, we're actually interested in the linearized space of fields. By default, define $\cC_l(X) = \c[\cC(X)]$; that is, $\cC_l(X)$ is -the vector space of finite +the vector space of finite linear combinations of fields on $X$. If $X$ has boundary, we of course fix a boundary condition: $\cC_l(X; a) = \c[\cC(X; a)]$. Thus the restriction (to boundary) maps are well defined because we never @@ -220,9 +214,9 @@ In some cases we don't linearize the default way; instead we take the spaces $\cC_l(X; a)$ to be part of the data for the system of fields. In particular, for fields based on linear $n$-category pictures we linearize as follows. -Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by +Define $\cC_l(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by obvious relations on 0-cell labels. -More specifically, let $L$ be a cell decomposition of $X$ +More specifically, let $L$ be a cell decomposition of $X$ and let $p$ be a 0-cell of $L$. Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that $\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. @@ -231,9 +225,9 @@ to infer the meaning of $\alpha_{\lambda c + d}$. Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. -\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; +\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; will do something similar below; in general, whenever a label lives in a linear -space we do something like this; ? say something about tensor +space we do something like this; ? say something about tensor product of all the linear label spaces? Yes:} For top dimensional ($n$-dimensional) manifolds, we linearize as follows. @@ -243,7 +237,7 @@ space determined by the labeling of the link of the 0-cell. (If the 0-cell were labeled, the label would live in this space.) We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). -We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the +We now define $\cC_l(X; a)$ to be the direct sum over all almost labelings of the above tensor products. @@ -251,12 +245,12 @@ \subsection{Local relations} Let $B^n$ denote the standard $n$-ball. -A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ +A {\it local relation} is a collection subspaces $U(B^n; c) \sub \cC_l(B^n; c)$ (for all $c \in \cC(\bd B^n)$) satisfying the following (three?) properties. -\nn{Roughly, these are (1) the local relations imply (extended) isotopy; +\nn{Roughly, these are (1) the local relations imply (extended) isotopy; (2) $U(B^n; \cdot)$ is an ideal w.r.t.\ gluing; and -(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). +(3) this ideal is generated by ``small" generators (contained in an open cover of $B^n$). See KW TQFT notes for details. Need to transfer details to here.} For maps into spaces, $U(B^n; c)$ is generated by things of the form $a-b \in \cC_l(B^n; c)$, @@ -292,7 +286,7 @@ In this section we will usually suppress boundary conditions on $X$ from the notation (e.g. write $\cC_l(X)$ instead of $\cC_l(X; c)$). -We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 +We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 submanifold of $X$, then $X \setmin Y$ implicitly means the closure $\overline{X \setmin Y}$. @@ -326,7 +320,7 @@ There is a map $\bd : \bc_1(X) \to \bc_0(X)$ which sends $(B, r, u)$ to $ru$, the linear combination of fields on $X$ obtained by gluing $r$ to $u$. -In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by +In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by just erasing the blob from the picture (but keeping the blob label $u$). @@ -334,7 +328,7 @@ is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. $\bc_2(X)$ is the space of all relations (redundancies) among the relations of $\bc_1(X)$. -More specifically, $\bc_2(X)$ is the space of all finite linear combinations of +More specifically, $\bc_2(X)$ is the space of all finite linear combinations of 2-blob diagrams (defined below), modulo the usual linear label relations. \nn{and also modulo blob reordering relations?} @@ -403,14 +397,14 @@ Let $x_a$ be the blob diagram with label $a$, and define $x_b$ and $x_c$ similarly. Then we impose the relation \eq{ - x_c = \lambda x_a + x_b . + x_c = \lambda x_a + x_b . } \nn{should do this in terms of direct sums of tensor products} Let $x$ and $x'$ be two blob diagrams which differ only by a permutation $\pi$ of their blob labelings. Then we impose the relation \eq{ - x = \sign(\pi) x' . + x = \sign(\pi) x' . } (Alert readers will have noticed that for $k=2$ our definition @@ -430,7 +424,7 @@ where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. Finally, define \eq{ - \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). + \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). } The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. Thus we have a chain complex. @@ -438,8 +432,8 @@ \nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} -\nn{TO DO: -expand definition to handle DGA and $A_\infty$ versions of $n$-categories; +\nn{TO DO: +expand definition to handle DGA and $A_\infty$ versions of $n$-categories; relations to Chas-Sullivan string stuff} @@ -451,7 +445,7 @@ \end{prop} \begin{proof} Given blob diagrams $b_1$ on $X$ and $b_2$ on $Y$, we can combine them -(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a +(putting the $b_1$ blobs before the $b_2$ blobs in the ordering) to get a blob diagram $(b_1, b_2)$ on $X \du Y$. Because of the blob reordering relations, all blob diagrams on $X \du Y$ arise this way. In the other direction, any blob diagram on $X\du Y$ is equal (up to sign) @@ -467,8 +461,8 @@ For the next proposition we will temporarily restore $n$-manifold boundary conditions to the notation. -Suppose that for all $c \in \cC(\bd B^n)$ -we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ +Suppose that for all $c \in \cC(\bd B^n)$ +we have a splitting $s: H_0(\bc_*(B^n, c)) \to \bc_0(B^n; c)$ of the quotient map $p: \bc_0(B^n; c) \to H_0(\bc_*(B^n, c))$. \nn{always the case if we're working over $\c$}. @@ -490,7 +484,7 @@ \nn{$x$ is a 0-blob diagram, i.e. $x \in \cC(B^n; c)$} \end{proof} -(Note that for the above proof to work, we need the linear label relations +(Note that for the above proof to work, we need the linear label relations for blob labels. Also we need to blob reordering relations (?).) @@ -525,7 +519,7 @@ \begin{prop} For fixed fields ($n$-cat), $\bc_*$ is a functor from the category -of $n$-manifolds and diffeomorphisms to the category of chain complexes and +of $n$-manifolds and diffeomorphisms to the category of chain complexes and (chain map) isomorphisms. \qed \end{prop} @@ -558,9 +552,9 @@ \begin{prop} There is a natural chain map \eq{ - \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). + \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). } -The sum is over all fields $a$ on $Y$ compatible at their +The sum is over all fields $a$ on $Y$ compatible at their ($n{-}2$-dimensional) boundaries with $c$. `Natural' means natural with respect to the actions of diffeomorphisms. \qed @@ -574,7 +568,7 @@ (Typically one of $X_1$ or $X_2$ is a disjoint union of balls.) For $x_i \in \bc_*(X_i)$, we introduce the notation \eq{ - x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . + x_1 \bullet x_2 \deq \gl(x_1 \otimes x_2) . } Note that we have resumed our habit of omitting boundary labels from the notation. @@ -589,17 +583,17 @@ \section{$n=1$ and Hochschild homology} In this section we analyze the blob complex in dimension $n=1$ -and find that for $S^1$ the homology of the blob complex is the +and find that for $S^1$ the homology of the blob complex is the Hochschild homology of the category (algebroid) that we started with. \nn{or maybe say here that the complexes are quasi-isomorphic? in general, should perhaps put more emphasis on the complexes and less on the homology.} Notation: $HB_i(X) = H_i(\bc_*(X))$. -Let us first note that there is no loss of generality in assuming that our system of +Let us first note that there is no loss of generality in assuming that our system of fields comes from a category. (Or maybe (???) there {\it is} a loss of generality. -Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be +Given any system of fields, $A(I; a, b) = \cC(I; a, b)/U(I; a, b)$ can be thought of as the morphisms of a 1-category $C$. More specifically, the objects of $C$ are $\cC(pt)$, the morphisms from $a$ to $b$ are $A(I; a, b)$, and composition is given by gluing. @@ -624,7 +618,7 @@ \begin{itemize} \item $\cC(pt) = \ob(C)$ . \item Let $R$ be a 1-manifold and $c \in \cC(\bd R)$. -Then an element of $\cC(R; c)$ is a collection of (transversely oriented) +Then an element of $\cC(R; c)$ is a collection of (transversely oriented) points in the interior of $R$, each labeled by a morphism of $C$. The intervals between the points are labeled by objects of $C$, consistent with @@ -635,12 +629,12 @@ the same way. \item For $x \in \mor(a, b)$ let $\chi(x) \in \cC(I; a, b)$ be the field with a single point (at some standard location) labeled by $x$. -Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the +Then the kernel of the evaluation map $U(I; a, b)$ is generated by things of the form $y - \chi(e(y))$. Thus we can, if we choose, restrict the blob twig labels to things of this form. \end{itemize} -We want to show that $HB_*(S^1)$ is naturally isomorphic to the +We want to show that $HB_*(S^1)$ is naturally isomorphic to the Hochschild homology of $C$. \nn{Or better that the complexes are homotopic or quasi-isomorphic.} @@ -691,13 +685,13 @@ First we show that $F_*(C\otimes C)$ is quasi-isomorphic to the 0-step complex $C$. -Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of +Let $F'_* \sub F_*(C\otimes C)$ be the subcomplex where the label of the point $*$ is $1 \otimes 1 \in C\otimes C$. We will show that the inclusion $i: F'_* \to F_*(C\otimes C)$ is a quasi-isomorphism. Fix a small $\ep > 0$. Let $B_\ep$ be the ball of radius $\ep$ around $* \in S^1$. -Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex +Let $F^\ep_* \sub F_*(C\otimes C)$ be the subcomplex generated by blob diagrams $b$ such that $B_\ep$ is either disjoint from or contained in each blob of $b$, and the two boundary points of $B_\ep$ are not labeled points of $b$. For a field (picture) $y$ on $B_\ep$, let $s_\ep(y)$ be the equivalent picture with~$*$ @@ -712,7 +706,7 @@ $x$ as a new twig blob, with label $y - s_\ep(y)$, where $y$ is the restriction of $x$ to $B_\ep$. If $*$ is contained in a twig blob $B$ with label $u = \sum z_i$, $j_\ep(x)$ is obtained as follows. Let $y_i$ be the restriction of $z_i$ to $B_\ep$. -Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, +Let $x_i$ be equal to $x$ outside of $B$, equal to $z_i$ on $B \setmin B_\ep$, and have an additional blob $B_\ep$ with label $y_i - s_\ep(y_i)$. Define $j_\ep(x) = \sum x_i$. \nn{need to check signs coming from blob complex differential} @@ -721,7 +715,7 @@ The key property of $j_\ep$ is \eq{ - \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , + \bd j_\ep + j_\ep \bd = \id - \sigma_\ep , } where $\sigma_\ep : F^\ep_* \to F^\ep_*$ is given by replacing the restriction $y$ of each field mentioned in $x \in F^\ep_*$ with $s_\ep(y)$. @@ -731,10 +725,10 @@ is a homotopy inverse to the inclusion $F'_* \to F_*(C\otimes C)$. One strategy would be to try to stitch together various $j_\ep$ for progressively smaller $\ep$ and show that $F'_*$ is homotopy equivalent to $F_*(C\otimes C)$. -Instead, we'll be less ambitious and just show that +Instead, we'll be less ambitious and just show that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. -If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have +If $x$ is a cycle in $F_*(C\otimes C)$, then for sufficiently small $\ep$ we have $x \in F_*^\ep$. (This is true for any chain in $F_*(C\otimes C)$, since chains are sums of finitely many blob diagrams.) @@ -743,7 +737,7 @@ If $y \in F_*(C\otimes C)$ and $\bd y = x \in F'_*$, then $y \in F^\ep_*$ for some $\ep$ and \eq{ - \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . + \bd y = \bd (\sigma_\ep(y) + j_\ep(x)) . } Since $\sigma_\ep(y) + j_\ep(x) \in F'$, it follows that the inclusion map is injective on homology. This completes the proof that $F'_*$ is quasi-isomorphic to $F_*(C\otimes C)$. @@ -769,7 +763,7 @@ Next we construct a degree 1 map (homotopy) $h: F'_* \to F'_*$ such that for all $x \in F'_*$ we have \eq{ - x - \bd h(x) - h(\bd x) \in F''_* . + x - \bd h(x) - h(\bd x) \in F''_* . } Since $F'_0 = F''_0$, we can take $h_0 = 0$. Let $x \in F'_1$, with single blob $B \sub S^1$. @@ -793,7 +787,7 @@ Finally, we show that $F''_*$ is contractible. \nn{need to also show that $H_0$ is the right thing; easy, but I won't do it now} Let $x$ be a cycle in $F''_*$. -The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a +The union of the supports of the diagrams in $x$ does not contain $*$, so there exists a ball $B \subset S^1$ containing the union of the supports and not containing $*$. Adding $B$ as a blob to $x$ gives a contraction. \nn{need to say something else in degree zero} @@ -813,11 +807,11 @@ * is a labeled point in $y$. Otherwise, define $s(y)$ to be the result of adding a label 1 (identity morphism) at *. Let $x \in \bc_*(S^1)$. -Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in +Let $s(x)$ be the result of replacing each field $y$ (containing *) mentioned in $x$ with $y$. It is easy to check that $s$ is a chain map and $s \circ i = \id$. -Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points +Let $G^\ep_* \sub \bc_*(S^1)$ be the subcomplex where there are no labeled points in a neighborhood $B_\ep$ of *, except perhaps *. Note that for any chain $x \in \bc_*(S^1)$, $x \in G^\ep_*$ for sufficiently small $\ep$. \nn{rest of argument goes similarly to above} @@ -833,8 +827,36 @@ Probably it's worth writing down an explicit map even if we don't need to.} +We can also describe explicitly a map from the standard Hochschild +complex to the blob complex on the circle. \nn{What properties does this +map have?} +\begin{figure}% +$$\mathfig{0.6}{barycentric/barycentric}$$ +\caption{The Hochschild chain $a \tensor b \tensor c$ is sent to +the sum of six blob $2$-chains, corresponding to a barycentric subdivision of a $2$-simplex.} +\label{fig:Hochschild-example}% +\end{figure} +As an example, Figure \ref{fig:Hochschild-example} shows the image of the Hochschild chain $a \tensor b \tensor c$. Only the $0$-cells are shown explicitly. +The edges marked $x, y$ and $z$ carry the $1$-chains +\begin{align*} +x & = \mathfig{0.1}{barycentric/ux} & u_x = \mathfig{0.1}{barycentric/ux_ca} - \mathfig{0.1}{barycentric/ux_c-a} \\ +y & = \mathfig{0.1}{barycentric/uy} & u_y = \mathfig{0.1}{barycentric/uy_cab} - \mathfig{0.1}{barycentric/uy_ca-b} \\ +z & = \mathfig{0.1}{barycentric/uz} & u_z = \mathfig{0.1}{barycentric/uz_c-a-b} - \mathfig{0.1}{barycentric/uz_cab} +\end{align*} +and the $2$-chain labelled $A$ is +\begin{equation*} +A = \mathfig{0.1}{barycentric/Ax}+\mathfig{0.1}{barycentric/Ay}. +\end{equation*} +Note that we then have +\begin{equation*} +\bdy A = x+y+z. +\end{equation*} + +In general, the Hochschild chain $\Tensor_{i=1}^n a_i$ is sent to the sum of $n!$ blob $(n-1)$-chains, indexed by permutations, +$$\phi\left(\Tensor_{i=1}^n a_i) = \sum_{\pi} \phi^\pi(a_1, \ldots, a_n)$$ +with ... \section{Action of $C_*(\Diff(X))$} \label{diffsect} @@ -849,16 +871,16 @@ \begin{prop} \label{CDprop} For each $n$-manifold $X$ there is a chain map \eq{ - e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . + e_X : CD_*(X) \otimes \bc_*(X) \to \bc_*(X) . } On $CD_0(X) \otimes \bc_*(X)$ it agrees with the obvious action of $\Diff(X)$ on $\bc_*(X)$ (Proposition (\ref{diff0prop})). For any splitting $X = X_1 \cup X_2$, the following diagram commutes \eq{ \xymatrix{ - CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ - CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) - \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & - \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} + CD_*(X) \otimes \bc_*(X) \ar[r]^{e_X} & \bc_*(X) \\ + CD_*(X_1) \otimes CD_*(X_2) \otimes \bc_*(X_1) \otimes \bc_*(X_2) + \ar@/_4ex/[r]_{e_{X_1} \otimes e_{X_2}} \ar[u]^{\gl \otimes \gl} & + \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl} } } Any other map satisfying the above two properties is homotopic to $e_X$. \end{prop} @@ -876,18 +898,18 @@ A $k$-parameter family of diffeomorphisms $f: P \times X \to X$ is {\it adapted to $\cU$} if there is a factorization \eq{ - P = P_1 \times \cdots \times P_m + P = P_1 \times \cdots \times P_m } (for some $m \le k$) and families of diffeomorphisms \eq{ - f_i : P_i \times X \to X + f_i : P_i \times X \to X } -such that +such that \begin{itemize} \item each $f_i(p, \cdot): X \to X$ is supported on some connected $V_i \sub X$; \item the $V_i$'s are mutually disjoint; -\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, +\item each $V_i$ is the union of at most $k_i$ of the $U_\alpha$'s, where $k_i = \dim(P_i)$; and \item $f(p, \cdot) = f_1(p_1, \cdot) \circ \cdots \circ f_m(p_m, \cdot) \circ g$ for all $p = (p_1, \ldots, p_m)$, for some fixed $g \in \Diff(X)$. @@ -904,12 +926,12 @@ \medskip -Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ +Let $B_1, \ldots, B_m$ be a collection of disjoint balls in $X$ (e.g.~the support of a blob diagram). We say that $f:P\times X\to X$ is {\it compatible} with $\{B_j\}$ if $f$ has support a disjoint collection of balls $D_i \sub X$ and for all $i$ and $j$ either $B_j \sub D_i$ or $B_j \cap D_i = \emptyset$. -A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, +A chain $x \in CD_k(X)$ is compatible with $\{B_j\}$ if it is a sum of singular cells, each of which is compatible. (Note that we could strengthen the definition of compatibility to incorporate a factorization condition, similar to the definition of ``adapted to" above. @@ -920,14 +942,14 @@ Then $x$ is homotopic (rel boundary) to some $x' \in CD_k(X)$ which is compatible with $\{B_j\}$. \end{cor} \begin{proof} -This will follow from Lemma \ref{extension_lemma} for +This will follow from Lemma \ref{extension_lemma} for appropriate choice of cover $\cU = \{U_\alpha\}$. Let $U_{\alpha_1}, \ldots, U_{\alpha_k}$ be any $k$ open sets of $\cU$, and let $V_1, \ldots, V_m$ be the connected components of $U_{\alpha_1}\cup\cdots\cup U_{\alpha_k}$. Choose $\cU$ fine enough so that there exist disjoint balls $B'_j \sup B_j$ such that for all $i$ and $j$ either $V_i \sub B'_j$ or $V_i \cap B'_j = \emptyset$. -Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, +Apply Lemma \ref{extension_lemma} first to each singular cell $f_i$ of $\bd x$, with the (compatible) support of $f_i$ in place of $X$. This insures that the resulting homotopy $h_i$ is compatible. Now apply Lemma \ref{extension_lemma} to $x + \sum h_i$. @@ -957,7 +979,7 @@ \nn{not sure what the best way to deal with boundary is; for now just give main argument, worry about boundary later} -Recall that we are given +Recall that we are given an open cover $\cU = \{U_\alpha\}$ and an $x \in CD_k(X)$ such that $\bd x$ is adapted to $\cU$. We must find a homotopy of $x$ (rel boundary) to some $x' \in CD_k(X)$ which is adapted to $\cU$. @@ -965,20 +987,20 @@ Let $\{r_\alpha : X \to [0,1]\}$ be a partition of unity for $\cU$. As a first approximation to the argument we will eventually make, let's replace $x$ -with a single singular cell +with a single singular cell \eq{ - f: P \times X \to X . + f: P \times X \to X . } Also, we'll ignore for now issues around $\bd P$. Our homotopy will have the form \eqar{ - F: I \times P \times X &\to& X \\ - (t, p, x) &\mapsto& f(u(t, p, x), x) + F: I \times P \times X &\to& X \\ + (t, p, x) &\mapsto& f(u(t, p, x), x) } for some function \eq{ - u : I \times P \times X \to P . + u : I \times P \times X \to P . } First we describe $u$, then we argue that it does what we want it to do. @@ -1007,16 +1029,16 @@ For $p \in D$ we define \eq{ - u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . + u(t, p, x) = (1-t)p + t \sum_\alpha r_\alpha(x) p_{c_\alpha} . } (Recall that $P$ is a single linear cell, so the weighted average of points of $P$ makes sense.) So far we have defined $u(t, p, x)$ when $p$ lies in a $k$-handle of $J$. -For handles of $J$ of index less than $k$, we will define $u$ to +For handles of $J$ of index less than $k$, we will define $u$ to interpolate between the values on $k$-handles defined above. -If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate +If $p$ lies in a $k{-}1$-handle $E$, let $\eta : E \to [0,1]$ be the normal coordinate of $E$. In particular, $\eta$ is equal to 0 or 1 only at the intersection of $E$ with a $k$-handle. @@ -1026,8 +1048,8 @@ adjacent to the $k{-}1$-cell corresponding to $E$. For $p \in E$, define \eq{ - u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} - + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . + u(t, p, x) = (1-t)p + t \left( \sum_{\alpha \ne \beta} r_\alpha(x) p_{c_\alpha} + + r_\beta(x) (\eta(p) q_1 + (1-\eta(p)) q_0) \right) . } In general, for $E$ a $k{-}j$-handle, there is a normal coordinate @@ -1040,10 +1062,10 @@ For each $\beta \in \cN$, let $\{q_{\beta i}\}$ be the set of points in $P$ associated to the aforementioned $k$-cells. Now define, for $p \in E$, \eq{ - u(t, p, x) = (1-t)p + t \left( - \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} - + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) - \right) . + u(t, p, x) = (1-t)p + t \left( + \sum_{\alpha \notin \cN} r_\alpha(x) p_{c_\alpha} + + \sum_{\beta \in \cN} r_\beta(x) \left( \sum_i \eta_{\beta i}(p) \cdot q_{\beta i} \right) + \right) . } Here $\eta_{\beta i}(p)$ is the weight given to $q_{\beta i}$ by the linear extension mentioned above. @@ -1062,8 +1084,8 @@ We must show that the derivative $\pd{F}{x}(t, p, x)$ is non-singular for all $(t, p, x)$. We have \eq{ -% \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . - \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . +% \pd{F}{x}(t, p, x) = \pd{f}{x}(u(t, p, x), x) + \pd{f}{p}(u(t, p, x), x) \pd{u}{x}(t, p, x) . + \pd{F}{x} = \pd{f}{x} + \pd{f}{p} \pd{u}{x} . } Since $f$ is a family of diffeomorphisms, $\pd{f}{x}$ is non-singular and \nn{bounded away from zero, or something like that}. @@ -1083,7 +1105,7 @@ This will complete the proof of the lemma. \nn{except for boundary issues and the `$P$ is a cell' assumption} -Let $j$ be the codimension of $D$. +Let $j$ be the codimension of $D$. (Or rather, the codimension of its corresponding cell. From now on we will not make a distinction between handle and corresponding cell.) Then $j = j_1 + \cdots + j_m$, $0 \le m \le k$, @@ -1110,7 +1132,7 @@ The singular 1-cell is supported on $U_\beta$, since $r_\beta = 0$ outside of this set. Next case: $j=2$, $m=1$, $j_1 = 2$. -This is similar to the previous case, except that the normal bundle is 2-dimensional instead of +This is similar to the previous case, except that the normal bundle is 2-dimensional instead of 1-dimensional. We have that $D \to \Diff(X)$ is a product of a constant singular $k{-}2$-cell and a 2-cell with support $U_\beta$. @@ -1136,15 +1158,15 @@ \section{$A_\infty$ action on the boundary} Let $Y$ be an $n{-}1$-manifold. -The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary +The collection of complexes $\{\bc_*(Y\times I; a, b)\}$, where $a, b \in \cC(Y)$ are boundary conditions on $\bd(Y\times I) = Y\times \{0\} \cup Y\times\{1\}$, has the structure of an $A_\infty$ category. Composition of morphisms (multiplication) depends of a choice of homeomorphism $I\cup I \cong I$. Given this choice, gluing gives a map \eq{ - \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) - \cong \bc_*(Y\times I; a, c) + \bc_*(Y\times I; a, b) \otimes \bc_*(Y\times I; b, c) \to \bc_*(Y\times (I\cup I); a, c) + \cong \bc_*(Y\times I; a, c) } Using (\ref{CDprop}) and the inclusion $\Diff(I) \sub \Diff(Y\times I)$ gives the various higher associators of the $A_\infty$ structure, more or less canonically. @@ -1155,7 +1177,7 @@ Similarly, if $Y \sub \bd X$, a choice of collaring homeomorphism $(Y\times I) \cup_Y X \cong X$ gives the collection of complexes $\bc_*(X; r, a)$ -(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the +(variable $a \in \cC(Y)$; fixed $r \in \cC(\bd X \setmin Y)$) the structure of a representation of the $A_\infty$ category $\{\bc_*(Y\times I; \cdot, \cdot)\}$. Again the higher associators come from the action of $\Diff(I)$ on a collar neighborhood of $Y$ in $X$. @@ -1176,14 +1198,14 @@ Gluing the two copies of $Y$ together we obtain a new $n$-manifold $X\sgl$. We wish to describe the blob complex of $X\sgl$ in terms of the blob complex of $X$. -More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, +More precisely, we want to describe $\bc_*(X\sgl; c\sgl)$, where $c\sgl \in \cC(\bd X\sgl)$, in terms of the collection $\{\bc_*(X; c, \cdot, \cdot)\}$, thought of as a representation of the $A_\infty$ category $\cA(Y\du-Y) \cong \cA(Y)\times \cA(Y)\op$. \begin{thm} $\bc_*(X\sgl; c\sgl)$ is quasi-isomorphic to the the self tensor product -of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. +of $\{\bc_*(X; c, \cdot, \cdot)\}$ over $\cA(Y)$. \end{thm} The proof will occupy the remainder of this section. @@ -1200,7 +1222,7 @@ \section{Extension to ...} -\nn{Need to let the input $n$-category $C$ be a graded thing +\nn{Need to let the input $n$-category $C$ be a graded thing (e.g.~DGA or $A_\infty$ $n$-category).} \nn{maybe this should be done earlier in the exposition? @@ -1233,6 +1255,3 @@ %$m: \bc_0(B^n; c, c') \to \mor(c, c')$. %If we regard $\mor(c, c')$ as a complex concentrated in degree 0, then this becomes a chain %map $m: \bc_*(B^n; c, c') \to \mor(c, c')$. - - - diff -r 4ef2f77a4652 -r 15e6335ff1d4 build.xml --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/build.xml Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,96 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/latex2pdf/README.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/latex2pdf/README.txt Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,3 @@ +*Create defontify.tex containing the latex you want. +*Run defontify.bat. +*Open nofonts.ps in Illustrator, and copy the pieces you want over to the .pdf you're working on. diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/latex2pdf/defontify.bat --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/latex2pdf/defontify.bat Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,4 @@ +latex defontify +dvips -R0 defontify +ps2pdf defontify.ps +gs -r9600 -sDEVICE=pswrite -dNOCACHE -sOutputFile=nofonts.ps -q -dbatch -dNOPAUSE defontify.pdf -c quit diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/latex2pdf/defontify.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/latex2pdf/defontify.tex Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,15 @@ +\documentclass{amsart} + +\newcommand{\pathtotrunk}{../../} +% \input{\pathtotrunk text/article_preamble.tex} + +\pagestyle{empty} + +\begin{document} +\thispagestyle{empty} + +\begin{align*} +abc bca cab ab bc ca a b c \\ +x y z A u_x u_y u_z +\end{align*} +\end{document} diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/Ax.pdf Binary file diagrams/pdf/barycentric/Ax.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/Ay.pdf Binary file diagrams/pdf/barycentric/Ay.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/barycentric.pdf Binary file diagrams/pdf/barycentric/barycentric.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/ux.pdf Binary file diagrams/pdf/barycentric/ux.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/ux_c-a.pdf Binary file diagrams/pdf/barycentric/ux_c-a.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/ux_ca.pdf Binary file diagrams/pdf/barycentric/ux_ca.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/uy.pdf Binary file diagrams/pdf/barycentric/uy.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/uy_ca-b.pdf Binary file diagrams/pdf/barycentric/uy_ca-b.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/uy_cab.pdf Binary file diagrams/pdf/barycentric/uy_cab.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/uz.pdf Binary file diagrams/pdf/barycentric/uz.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/uz_c-a-b.pdf Binary file diagrams/pdf/barycentric/uz_c-a-b.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/pdf/barycentric/uz_cab.pdf Binary file diagrams/pdf/barycentric/uz_cab.pdf has changed diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/README.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/README.txt Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,11 @@ +The scripts in this directory automatically build .eps files out of .pdf +files. You can ignore it if you're using pdflatex. + +find_all_diagrams attempts to determine which diagrams are being +used, and creates the file diagrams.list + +stripall creates small .eps files from the .pdf files named in +diagrams.list and extra_diagrams.list + +If find_all_diagrams isn't finding everything you need, add things +by hand to extra_diagrams.list diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/diagrams.list --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/diagrams.list Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,22 @@ +quasitriangular +kauffmanbracket +biggerbraiding +SampleTangle +RemovingSpaghettiO +TraceTangle +Multiplication +TL3 +TLExample +Eis +AnnularConsequences +TwoRs +TwoRsInTL +TwoRs +TwoRs +TwoRs +TwoRsInTL +TwoRsInTL +TwoRsInTL +SampleTangle +translation/rectangular +translation/composition diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/extra_diagrams.list --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/extra_diagrams.list Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ + diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/find_all_diagrams.bat --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/find_all_diagrams.bat Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +bash find_all_diagrams.sh diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/find_all_diagrams.sh --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/find_all_diagrams.sh Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,9 @@ +rm diagrams.list +for source in `ls ../../text/*.tex`; + do + ./find_diagrams.sh $source +done; +./find_diagrams.sh ../../sandbox.tex +./find_diagrams.sh ../../todolist.tex +./find_diagrams.sh ../latex2pdf/defontify.tex +cat diagrams.list diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/find_diagrams.sh --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/find_diagrams.sh Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,3 @@ +grep -o '\\\mathfig{[0-9\. ]*}{[a-zA-z0-9\/_-]*}' $1 | sed -e 's/\\\mathfig{.*{\(.*\)/\1/' | sed -e 's/}//g' >> diagrams.list +grep -o '\\\placefig{[0-9\. ]*}{[a-zA-z0-9\/_-]*}' $1 | sed -e 's/\\\placefig{.*{\(.*\)/\1/' | sed -e 's/}//g' >> diagrams.list +grep -o '\\\rotatemathfig{ [0-9\.]*}{[0-9\.-]*}{[a-zA-z0-9\/_-]*}' $1 | sed -e 's/\\\rotatemathfig{.*{\(.*\)/\1/' | sed -e 's/}//g' >> diagrams.list diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/stripall.bat --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/stripall.bat Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,1 @@ +bash stripall.sh diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/stripall.sh --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/stripall.sh Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,19 @@ +for diagram in `cat diagrams.list`; + do + if [ ../pdf/$diagram.pdf -nt ../eps/$diagram.eps ]; then + echo "*** stripping $diagram"; + ./strippdf.sh $diagram; + else + echo "$diagram is up to date" + fi +done; +dos2unix extra_diagrams.list +for diagram in `cat extra_diagrams.list`; + do + if [ ../pdf/$diagram.pdf -nt ../eps/$diagram.eps ]; then + echo "*** stripping $diagram"; + ./strippdf.sh $diagram; + else + echo "$diagram is up to date" + fi +done; diff -r 4ef2f77a4652 -r 15e6335ff1d4 diagrams/scripts/strippdf.sh --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diagrams/scripts/strippdf.sh Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,4 @@ +gs -r2400 -dBATCH -dNOPAUSE -dSAFER -q -dNOCACHE -sDEVICE=epswrite -sOutputFile=temp1.eps ../pdf/$1.pdf +sed -e '/CreationDate/d' temp1.eps > temp2.eps +rm temp1.eps +mv temp2.eps ../eps/$1.eps diff -r 4ef2f77a4652 -r 15e6335ff1d4 preamble.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/preamble.tex Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,189 @@ +%auto-ignore +%this ensures the arxiv doesn't try to start TeXing here. + +\usepackage{amsmath,amssymb,amsfonts} +\usepackage{ifpdf} + +%\ifpdf +%\usepackage[pdftex,all,color]{xy} +%\else +\usepackage[all,color]{xy} +%\fi + +\SelectTips{cm}{} +% This may speed up compilation of complex documents with many xymatrices. +%\CompileMatrices + +% ---------------------------------------------------------------- +\vfuzz2pt % Don't report over-full v-boxes if over-edge is small +\hfuzz2pt % Don't report over-full h-boxes if over-edge is small +% ---------------------------------------------------------------- + +% diagrams ------------------------------------------------------- +% figures --------------------------------------------------------- +%%% borrowed from Dror's cobordisms paper, use this to include eps or pdf graphics. +\ifpdf +\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/pdf/} +\else +\newcommand{\pathtodiagrams}{\pathtotrunk diagrams/eps/} +\fi + +\newcommand{\mathfig}[2]{{\hspace{-3pt}\begin{array}{c}% + \raisebox{-2.5pt}{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}% +\end{array}\hspace{-3pt}}} +\newcommand{\reflectmathfig}[2]{{\hspace{-3pt}\begin{array}{c}% + \raisebox{-2.5pt}{\reflectbox{\includegraphics[width=#1\textwidth]{\pathtodiagrams #2}}}% +\end{array}\hspace{-3pt}}} +\newcommand{\rotatemathfig}[3]{{\hspace{-3pt}\begin{array}{c}% + \raisebox{-2.5pt}{\rotatebox{#2}{\includegraphics[height=#1\textwidth]{\pathtodiagrams #3}}}% +\end{array}\hspace{-3pt}}} +\newcommand{\placefig}[2]{\includegraphics[width=#1\linewidth]{\pathtodiagrams #2}} + +\ifpdf +\usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref} +\else +\usepackage[dvips,plainpages=false,hypertexnames=false]{hyperref} +\fi +\newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} +\newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} +\newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} + + +% THEOREMS ------------------------------------------------------- +\theoremstyle{plain} +\newtheorem*{fact}{Fact} +\newtheorem{prop}{Proposition}[section] +\newtheorem{conj}[prop]{Conjecture} +\newtheorem{thm}[prop]{Theorem} +\newtheorem{lem}[prop]{Lemma} +\newtheorem{lemma}[prop]{Lemma} +\newtheorem{cor}[prop]{Corollary} +\newtheorem*{cor*}{Corollary} +\newtheorem*{exc}{Exercise} +%\theoremstyle{definition} +\newtheorem{defn}[prop]{Definition} % numbered definition +\newtheorem*{defn*}{Definition} % unnumbered definition +\newtheorem{question}{Question} +\newenvironment{rem}{\noindent\textsl{Remark.}}{} % perhaps looks better than rem above? +\numberwithin{equation}{section} +%\numberwithin{figure}{section} + +% Marginal notes in draft mode ----------------------------------- +\newcommand{\scott}[1]{\stepcounter{comment}{{\color{blue} $\star^{(\arabic{comment})}$}}\marginpar{\color{blue} $\star^{(\arabic{comment})}$ \usefont{T1}{scott}{m}{n} #1 --S}} % draft mode +\newcommand{\ari}[1]{\stepcounter{comment}{\color{red} $\star^{(\arabic{comment})}$}\marginpar{\color{red} $\star^{(\arabic{comment})}$ #1 --A}} % draft mode +\newcommand{\comment}[1]{\stepcounter{comment}$\star^{(\arabic{comment})}$\marginpar{\tiny $\star^{(\arabic{comment})}$ #1}} % draft mode +\newcounter{comment} +\newcommand{\noop}[1]{} +\newcommand{\todo}[1]{\textbf{TODO: #1}} + +% \mathrlap -- a horizontal \smash-------------------------------- +% For comparison, the existing overlap macros: +% \def\llap#1{\hbox to 0pt{\hss#1}} +% \def\rlap#1{\hbox to 0pt{#1\hss}} +\def\clap#1{\hbox to 0pt{\hss#1\hss}} +\def\mathllap{\mathpalette\mathllapinternal} +\def\mathrlap{\mathpalette\mathrlapinternal} +\def\mathclap{\mathpalette\mathclapinternal} +\def\mathllapinternal#1#2{% +\llap{$\mathsurround=0pt#1{#2}$}} +\def\mathrlapinternal#1#2{% +\rlap{$\mathsurround=0pt#1{#2}$}} +\def\mathclapinternal#1#2{% +\clap{$\mathsurround=0pt#1{#2}$}} + +% MATH ----------------------------------------------------------- +\newcommand{\Natural}{\mathbb N} +\newcommand{\Integer}{\mathbb Z} +\newcommand{\Rational}{\mathbb Q} +\newcommand{\Real}{\mathbb R} +\newcommand{\Complex}{\mathbb C} +\newcommand{\Field}{\mathbb F} + +\newcommand{\Id}{\boldsymbol{1}} +\renewcommand{\imath}{\mathfrak{i}} +\renewcommand{\jmath}{\mathfrak{j}} + +\newcommand{\qRing}{\Integer[q,q^{-1}]} +\newcommand{\qMod}{\qRing-\operatorname{Mod}} +\newcommand{\ZMod}{\Integer-\operatorname{Mod}} + +\newcommand{\To}{\rightarrow} +\newcommand{\Into}{\hookrightarrow} +\newcommand{\Onto}{\mapsto} +\newcommand{\Iso}{\cong} +\newcommand{\ActsOn}{\circlearrowright} + +\newcommand{\htpy}{\simeq} + +\newcommand{\restrict}[2]{#1{}_{\mid #2}{}} +\newcommand{\set}[1]{\left\{#1\right\}} +\newcommand{\setc}[2]{\left\{#1 \;\left| \; #2 \right. \right\}} +\newcommand{\relations}[2]{\left<#1 \;\left| \; #2 \right. \right>} +\newcommand{\cone}[3]{C\left(#1 \xrightarrow{#2} #3\right)} +\newcommand{\pairing}[2]{\left\langle#1 ,#2 \right\rangle} + +\newcommand{\card}[1]{\sharp{#1}} + +\newcommand{\bdy}{\partial} +\newcommand{\compose}{\circ} +\newcommand{\eset}{\emptyset} + +\newcommand{\Cat}{\mathcal{C}} + +\newcommand{\psmallmatrix}[1]{\left(\begin{smallmatrix} #1 \end{smallmatrix}\right)} + +\newcommand{\qiq}[2]{[#1]_{#2}} +\newcommand{\qi}[1]{\qiq{#1}{q}} +\newcommand{\qdim}{\operatorname{dim_q}} + +\newcommand{\directSum}{\oplus} +\newcommand{\DirectSum}{\bigoplus} +\newcommand{\tensor}{\otimes} +\newcommand{\Tensor}{\bigotimes} + +\newcommand{\db}[1]{\left(\left(#1\right)\right)} + +\newcommand{\su}[1]{\mathfrak{su}_{#1}} +\newcommand{\csl}[1]{\mathfrak{sl}_{#1}} +\newcommand{\uqsl}[1]{U_q\left(\csl{#1}\right)} + +\newcommand{\Cobl}{{\mathcal Cob}_{/l}} +\newcommand{\Cob}[1]{{\mathcal Cob}\left(\su{#1}\right)} +\newcommand{\Kom}[1]{\operatorname{Kom}\left(#1\right)} + +\newcommand{\Mat}[1]{\mathbf{Mat}\left(#1\right)} +\newcommand{\Kar}[1]{\mathbf{Kar}\left(#1\right)} +\newcommand{\Inv}[1]{\operatorname{Inv}\left(#1\right)} +\newcommand{\Hom}[3]{\operatorname{Hom}_{#1}\left(#2,#3\right)} +\newcommand{\End}[1]{\operatorname{End}\left(#1\right)} + +\newcommand{\Gr}[2]{\text{Gr}(#1 \subset #2)} +\newcommand{\Flag}[3]{\text{Flag}(#1 \subset #2 \subset #3)} + +\def\llbracket{\left[\!\!\left[} +\def\rrbracket{\right]\!\!\right]} +\newcommand{\Kh}[1]{\llbracket#1\rrbracket} +\newcommand{\mirror}[1]{\overline{#1}}% + +\newcommand{\Tangles}{{\mathbf{Oriented Tangles}}} +\newcommand{\Spider}[1]{{\mathbf{Spider}\left(#1\right)}} +\newcommand{\TL}{\mathcal{TL}} +\newcommand{\Foam}[1]{\mathbf{Foam}\left(#1\right)} + +\newcommand{\directSumStack}[2]{{\begin{matrix}#1 \\ \DirectSum \\#2\end{matrix}}} +\newcommand{\directSumStackThree}[3]{{\begin{matrix}#1 \\ \DirectSum \\#2 \\ \DirectSum \\#3\end{matrix}}} + +\newcommand{\grading}[1]{{\color{blue}\{#1\}}} +\newcommand{\shift}[1]{\left[#1\right]} + +\newenvironment{narrow}[2]{% +\vspace{-0.4cm}% horrible hack, by scott % this only seems to be appropriate in beamer mode... +\begin{list}{}{% +\setlength{\topsep}{0pt}% +\setlength{\leftmargin}{#1}% +\setlength{\rightmargin}{#2}% +\setlength{\listparindent}{\parindent}% +\setlength{\itemindent}{\parindent}% +\setlength{\parsep}{\parskip}}% +\item[]}{\end{list}} +% ---------------------------------------------------------------- diff -r 4ef2f77a4652 -r 15e6335ff1d4 text/article_preamble.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/text/article_preamble.tex Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,29 @@ +%auto-ignore +%this ensures the arxiv doesn't try to start TeXing here. + +\input{\pathtotrunk preamble.tex} + +\usepackage{breakurl} + +\ifpdf +\usepackage[pdftex]{graphicx} +\else +\usepackage[dvips]{graphicx} +\fi + +\usepackage{color} + +% This switches fonts to the Palatino family. +\renewcommand{\familydefault}{ppl} + +%%% futzing with margins following Dror (from Karoubi) +%\marginparwidth 0pt% +%\marginparsep 0pt + +\textwidth 5.5in% +\textheight 9.0in% +\oddsidemargin 12pt% +\evensidemargin 12pt + +\topmargin -.6in% +\headsep .5in diff -r 4ef2f77a4652 -r 15e6335ff1d4 text/top_matter.tex --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/text/top_matter.tex Thu Apr 24 02:56:34 2008 +0000 @@ -0,0 +1,22 @@ +\title{Blob Homology} + +\author{Scott~Morrison} +\address{ +}% +\email{scott@tqft.net} \urladdr{http://tqft.net/} + +\author{Kevin~Walker} +\address{ +}% +\email{kevin@canyon23.net} \urladdr{http://canyon23.net/} + + +\date{ + First edition: the mysterious future + This edition: \today. +} + +%\primaryclass{57M25} \secondaryclass{57M27; 57Q45} +%\keywords{ + +%}