\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{mathrsfs}
\usepackage{latexsym}
\usepackage[all]{xy}
\setlength{\parindent}{0pt}
\setlength{\parskip}{2ex}
\addtolength{\hoffset}{-1cm}
\addtolength{\textwidth}{2cm}
\author{Gabriel C. Drummond-Cole}
\newtheorem{lemma}{Lemma}
\newtheorem{exercise}{Exercise}
\newtheorem{prop}{Proposition}
\newtheorem{example}{Example}
\newtheorem{defi}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}
\newtheorem{remark}{Remark}
\newtheorem{asdefi}{Analysis Situs Definition}
\newcommand{\boundary}{\delta}
\newcommand{\interior}{\ ^\circ}
\newcommand{\percent}{\%}
\newcommand{\transpose}{t}
\newcommand{\transversal}{\pitchfork}
\newcommand{\wh}{[unintelligible]}
\newcommand{\cC}{\mathbb{C}}
\newcommand{\Dd}{\mathscr{D}}
\newcommand{\eE}{\mathbb{E}}
\newcommand{\Ff}{\mathscr{F}}
\newcommand{\hH}{\mathbb{H}}
\newcommand{\Mm}{\mathscr{M}}
\newcommand{\Nn}{\mathscr{N}}
\newcommand{\nN}{\mathbb{N}}
\newcommand{\qQ}{\mathbb{Q}}
\newcommand{\rR}{\mathbb{R}}
\newcommand{\Ss}{\mathscr{S}}
\newcommand{\Vv}{\mathscr{V}}
\newcommand{\Xx}{\mathscr{X}}
\newcommand{\zZ}{\mathbb{Z}}
\newcommand{\la}[1]{\mathfrak{#1}}
\newcommand{\injects}{\hookrightarrow}
\newtheorem{conj}{Conjecture}

\title{Dennis Seminar\\February 7, 2007}
\begin{document}
\maketitle
Plan:
\begin{enumerate}\item Main example from topology over $R=\zZ[\pi,X]$
\item Theorem:\begin{enumerate}\item There is a free version \item lifting maps\end{enumerate}
\item Theorem re quasiisomorphism of subdivision
\item Pair subdivision
\item Dual cell decomposition and Poincar\'e duality
\end{enumerate}
I'm going to talk about some elementary things.  The overall idea is a paradigm, that a geometrical object like a space is like a chain complex, and the paradigm goes both ways, chain complexes behave like geometric objects.  That's the idea I wanted to push.  We've defined chain complexes and mentioned that they have homology groups.  These are strings of things, and the boundary squared is zero, and these are left $R$-modules.  We have the notion of maps and chain homotopies, $f-g=dA+Ad.$  We also have the notion of quasiisomorphism, you have maps each way and $fg\cong id, gf\cong id.$  This is a notion of equivalence, an important one.  This is more useful than strict isomorphism.  We observed that two maps, a chain mapping gives you an induced morphism on homology.  If two maps are chain homotopic, then $fz-gz$ is the boundary of $Az.$  The induced maps on homology are isomorphisms if you have a quasiisomorphism, and we wanted to ask the converse.  We're in the middle of analyzing that question, and we'll treat it with part two, but before that I wanted to treat the main example.  I mentioned the Eilenberg singular complex, but suppose you have a space divided up into cells, and there's the notion of a regular cell complex, which means the space is broken up into, each cell is equivalent to a closed ball.  This is technical to state but let's just say it's homeomorphic to a ball (but I want something slightly better).  It's not quite known that the better thing I want follows from being homeomorphic.  Right now a regular cell will mean that the closure of every cell  is homeomorphic to a closed ball.

[Me: Of the same dimension?]

Yes.  Because I want to kill you right now, that's a compliment.  You don't want a triangle where you identify the edges.  But you could take a barycentric subdivision, that would be regular.  So the first thing is, you arbitrarily choose an orientation of every cell, and then the chain complex is the free Abelian group on the cells of the appropriate dimensions.    Then the boundary is just the boundary, with signs according to whether the orientation of a boundary cell and the induced orientation from the original cell agree.

Pick a basepoint.  Now you can lift paths to the universal cover by choosing a lift of the basepoint.  I want to be lifting the cell decomposition.  Now for every cell you choose a homotopy class of paths.  Why don't you try to choose them coherently?  If the space is not simply connected, then it would not be coherent when you came back.  Now take the free $\zZ[\pi_1 X]$-module generated by the cells of $X$ decorated by an orientation and a homotopy class of path to the base point.  Now we get the boundary map as follows.  The boundary of a cell takes signs from orientation as before.  I have all of these paths, I go to the big cell from the basepoint and then return along the path of the smaller cell.  So when I write down the boundary of a cell, there are coefficients and signs times the faces.  You can think of this geometrically.  This is linear combinations of expressions $\pm g_i\text{cell}_i.$  Those are the elements here.  So this cell comes with a tail.  You can think geometrically of multiplying by a group element as being that cell with a different tail.  The elements are all the cells will all the possible tails.  This is just the universal cover.

Here we have a nice finite type complex of free $R$-modules.  This is an interesting example of a chain complex over a ring.  You can just take homology and get the homology of the universal cover.

If you take $C_\cdot,$ notice there is this natural augmentation $\zZ[\pi_1]\to \zZ.$

[Too tired.]

That's a construction from a geometrical object to a chain complex.  Suppose you have a map inducing an isomorphism hon  homology?
\begin{exercise}
Suppose that $D$ is the free thing that maps to $A$ and $\tilde{A}$ maps to $A$ and induces an isomorphism on homology.  Then there is an $\tilde{f}$ so that $\pi \tilde{f}-f=\boundary h+h\boundary.$
\end{exercise}
I'd also be happy to hear a proof of the free resolution without the degree lower bound.

\section{hypoelliptic Dirac Operator, Bismut}
This means this is less than elliptic, and so for us this means it is as good as elliptic, but for deeper analytic reasons.  I gave a talk last year on de Rham theory where the idea was that there should be a deformation of classical Hodge theory which should interpolate between the classical Hodge operator at zero and at infinity the geodesic flow or any nondegenerate Hamiltonian system on $T^*X$ by a family of pseudo-Laplacians.  They are hypoelliptic, second order, and when $b$ tends to zero, these operators converge to the Hodge Laplacian, and when $b\to\infty$ to the generator of the given Hamiltonian system.  We tend to think that ellipticity is stable, if you deform something elliptic you get something elliptic.  So in this context, it's an extremely unstable property.  This conceptually can become possible if we understand that the space on which this acts is always $T^*X$ except when $b=0$ when $T^*X$ collapses to $X.$

This picture was trying to make sense of the Witten deformation of $\box^{RM}.$  You want to interpolate between Hodge theory and MOorse theory.  This piture is wrong since there is no functional, but this indicates \wh

\section{Terilla}
Recall that $L=(V,d,[\,,\,])$ is a differential graded Lie algebra.  The Maurer Cartan equation is that $d\gamma+\frac{1}{2}[\gamma,\gamma]=0.$  Then $d_\gamma=d+[\gamma,\cdot]$ satisfies $d_\gamma^2=0.$  Now $\gamma\in V^1\otimes B, B$ is a parameter ring, $(V\otimes B, d_\gamma, [\,,\,])\to (V,d,[\,,\,])$

$B$ is augmented, commutative associatiative, unital over $k,$ $B\to k.$  A good class of examples is that $B$ is a local ring with one maximal ideal $m_B$ with the residue field $B/m_B$ equal to $k$ and $m_B$ finite dimensional.  So $B\cong k\oplus m_B\to k.$

\begin{defi}$\Aa$ is the category of Artin local $k$-algebras, with objects rings such $B$ and morphisms maps $B\to B',m_B\to m_{B'},$ the identity on $k.$  Given a differential graded Lie algebra one has a covariant funtor $Def_L:\Aa\to \text{Sets}$ which takes $$B\mapsto \{\gamma\in V^1\otimes m_B:d\gamma+\frac{1}{2}[\gamma,\gamma]=0\}/Exp(V^0\otimes m_B)$$
For every deformation problem there is a differential graded Lie algebra governing them.  There should be a concept that these are deformations with parameters and this is functorial.  This means that deformations of $X$ over a parameter ring $B$ up to equivalence is equivalent to $Def_L.$

If $A$ is an associative algebra over $k,$ then a deformation of $A$ over $B$ is an associative $B$-algebra structure $\mu:(A\otimes B)\otimes_B (A\otimes B)\to A\otimes B$ such that the map $A\otimes B\to A$ via augmentation is an algebra map.  This will be a covariant functor on parameter rings.  In this example, $\mu$ and $\mu'$ are equivalent if as $B$-algebras they are isomorphic.  So here this $L$ is $Hoch(A,A).$  For complex structures this is a Lie structure using holomorphic vector fields.

There will be a sequence of enhancements to this definition.  So first, pass from $\Aa$ to $\Aa^\zZ$ which are $\zZ$-graded algebras, and you can define $$Def^\zZ_L(B)=\{\gamma\in (V\otimes m_B)^1: d\gamma+\frac{1}{2}[\gamma,\gamma]=0\}$$  This brings in the entire Lie algebra, whereas before only the degree zero and one part were involved.

If you look at associative algebras again, deformations correspond to deformations of the Hochschild complex.  What does $Def^\zZ_L$ correspond to?  Deformations not in the category of associative but in the $A_\infty$ category.  One should have started with $A$ an $A_\infty$ structure and done the $\zZ$ graded version.  In the complex structure version of this, it's not clear geometrically what the deformations stand for, the right answer should include non-commutative manifolds, this already feels very quantum, and this is coming from a very small change in the discussion.

A moment ago I thought I wanted to say something.  What are some properties one could study?  Let's talk about infinitessimal deformations.  So take as an example $\epsilon=k[t]/t^2.$  So $\epsilon=k\oplus tk,$ with $t^2=0.$  It turns  out that $Def_L(\epsilon),$ by definition, these are equal to $\gamma\otimes m_\epsilon$ such that $d_\gamma+\frac{1}{2}[\gamma,\gamma]=0,$ so that $\gamma=\alpha\otimes t,$ so that $$d\gamma+\frac{1}{2}[\gamma,\gamma]=(d\alpha)t+\frac{1}{2}[\alpha,\alpha]t^2
$$
So that $d\alpha=0.$  These are equivalent if and only if $\alpha-\alpha'$ is a boundary.  So $Def_L(\epsilon)\cong H^1(L).$  

So $\epsilon_n$ is defined as $k[t]/t^2$ with degree of $t$ equal to $1-n.$  Then $Def_L^\zZ(\epsilon_n)\cong H^n(L),$ so that $\bigoplus_n Def_L^\zZ(\epsilon_n)\cong H^\cdot (L).$

There's already something intereseting to say.  If I look at solutions, I get a vector space, and that's a uniform situation.  In general, if $F:\Aa,\Aa^\zZ\to\text{Sets}$ and $F$ satisfies mild hypotheses, then $F(\epsilon)$ or $F(\epsilon_n)$ is a vector space.  In complex structure, Kodaira Spencer, infinitessimal deformations correspond to Cech cocycles.

\begin{defi} $\oplus_n F(\epsilon_n)$ is the tangent space to the functor $F$\end{defi}
\begin{defi}
A functor $F:\Aa\to \text{Sets}$ is representable by a pair $(R,\Gamma)$ where $R\in \widehat{\Aa^\zZ}$ and $\Gamma\in F(R).$  Then for all $B\in \Aa^\zZ$ and all $\gamma\in F(B)$ there exists a change in parameter rings $tau:R\to B$ so that $F(\tau):\Gamma\mapsto \gamma.$  In the context of Maurer Cartan, for differential graded Lie algebras there exists a versal parameter ring $R$ and a versal solution to the Maurer Cartan equation $\Gamma\in V\otimes m_R$ so that, well, think of $C^\infty(M)=R.$  You have a basepoint that represents your original structure or the zero solution to the Maurer Cartan equation, and then $p$ represents a particular deformation, I can pull back, there exists a map of spaces so that this deformation pulls back to a specified one in some other setting.

So in this case, one has the following situation, that the tangent space to $M$ at $0$ is isomorphic to the tangent space of the functor, $T_0(M)\cong F(\epsilon),$ in the situation $R=C^\infty(M)$ and $F$ is represented by $R.$
\end{defi}
\begin{thm}
If $F$ is smooth and representable, that means that $Spec(R)$ is a smooth space, then $R$ is a power series ring on the dual of the tangent space.  For differential graded Lie algebras, for $Def_L$ this means that $R=k[[t^1,\ldots, t^n]]$ where $t^i$ is a dual basis for $H(L).$\end{thm}
So this just means that you have $\Gamma=\sum t^\alpha \gamma_\alpha+t^\alpha t^\beta\gamma_{\alpha\beta}+\ldots$ for $\{\gamma_\alpha}$ a basis for $H(L).$  The equation tells us that $d\gamma_{\alpha\beta}=-[\gamma_\alpha,\gamma_\beta].$

I said I was going to say something about $L_\infty$-algebras.  So $\Gamma$ defines an $L_\infty$ homomorphism from the homology of $L$ into $L,$ so $(H,0,0)\to L=(V,d,[\,,\,]$  Here this means there are a sequence of maps $H^{\otimes j}\to L.$  The first is given by $[\gamma_\alpha]\to \gamma_\alpha.$  The second is given by $[\gamma_\alpha]\otimes [\gamma_\beta]\to \gamma_{\alpha\beta}.$

In the right context you always have representability, you get an $L_\infty$ structure on $(H,0,0)$ and a map to $L.$

Let me talk now about $L_\infty.$  We looked at these parameter rings.  You picked things to have good functorial properties under change of ring.  But you also want good properties under change of Lie algebra.

What is the $L_\infty$-algebra?  If I have $L\to L'$ then $Def_L(B)\to Def_{L'}(B).$  This is, you once told me that deformation theory was like algebraic topology.  I'm assuming homology is bounded on both sides so it deoesn't matter which direction the arrows go.

If $L,L'$ are quasiisomorphic as differential graded Lie algebras, as $Lie_\infty$ algebras, then \wh.  You want to define $Def_L$ for $L_\infty$ algebaras, replacing the exponential with \wh.
\begin{thm} If $L,L'$ are differential graded Lie algebras are quasi-isomorphcs as $L_\infty$ algebras then $Def_L$ and $\def_{L'}$ are canonically identified

Let me make a definition of $L_\infty.$  You have a bracket, doing it twice gives zero.  You extend the bracket to $SV\to SV.$  I really mean $S(V[1]).$  Shift the degree of $V$ by one and then take the graded symmetric coalgebra on it.  So $$a_1\wedge\ldots \wedge a_n=\sum_{r,s\ge 1}\epsilon a_{i_1}\wedge\ldots\wedge a_{i_r} \otimes (a_{j_1}\wedge\ldots\wedge a_{j_s})$$

So I want to extend the bracket to be a coderivation and so that this squares to zero.  You get all of these maps be on equal footing, so that, say, $(\tilde{d}+\widetilde{[\,,\,]})^2=0.$
\begin{defi} An $L_infty$ algebar is $(V,D)$ where $V$ is a graded vector space and $D$ is a coderivation of degree one and square zero.
\end{defi}

A map $L\to L'$ that gives an isomorphism at the level of homology, I could invert if these were vector spaces.  I can't do that for differential graded Lie algebra maps.  The best I can do is find an $L_\infty$ inverse map.  So this is the right category to talk about deformations.  This is the ``classical theory''  So the situation is that problems from physics involve $\hbar.$  Apply this idea to physical problems.  It is by carefully controlling $\hbar$ that gives you a way to correct the product so that the BV cohomology has an associative product in it.



\end{document}