diff --git a/thesis/images/Sullivan_Lifting.png b/thesis/images/Sullivan_Lifting.png new file mode 100644 index 0000000..16b3af7 Binary files /dev/null and b/thesis/images/Sullivan_Lifting.png differ diff --git a/thesis/notes/Algebra.tex b/thesis/notes/Algebra.tex index 1c6b2b5..8b76544 100644 --- a/thesis/notes/Algebra.tex +++ b/thesis/notes/Algebra.tex @@ -82,6 +82,7 @@ Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i = \todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary cdga's.} + \subsection{Homology} Whenever we have a differential graded module we have $d \circ d = 0$, or put in other words: the image of $d$ is a submodule of the kernel of $d$. The quotient of the two graded modules will be of interest. @@ -107,10 +108,32 @@ For differential graded algebras we can consider the (co)homology by forgetting Note that taking homology of a differential graded module (or algebra) is functorial. Whenever a map $f: M \to N$ of differential graded modules (or algebras) induces an isomorphism on homology, we say that $f$ is a \emph{quasi isomorphism}. +\begin{definition} + Let $M$ be a graded module. We say that $M$ is $n$-reduced if $M_i = 0$ for all $i \leq n$. Similarly we say that a graded algebra $A$ is $n$-reduced if $A_i = 0$ for all $1 \leq i \leq n$ and $\eta: \k \tot{\iso} A_0$. + + Let $(M, d)$ be a chain complex (or algebra). We say that $M$ is $n$-connected if $H(M)$ is $n$-reduced as graded module (resp. algebra). Similarly for cochain complexes. +\end{definition} + + +\subsection{Classical results} + +We will give some classical known results of algebraic topology or homological algebra. Proofs of these theorems can be found in many places. \todo{cite at least 1 place} + +\begin{theorem} + (Universal coefficient theorem) Let $C$ be a chain complex and $A$ an abelian group, then there are natural short exact sequences for each $n$: + $$ 0 \to H_n(C) \tensor A \to H_n(C \tensor A) \to Tor(H_{n-1}(C), A) \to 0 $$ + $$ 0 \to Ext(H_{n-1}(C), A) \to H^n(\Hom(C, A)) \to \Hom(H_n(C), A) \to 0 $$ +\end{theorem} + +The first statement generalizes to a theorem where $A$ is a chain complex itself. When choosing to work over a field the torsion will vanish and the exactness will induce an isomorphism. This is (one formulation of) the Künneth theorem. + +\begin{theorem} + (Künneth) Assume that $\k$ is a field and let $C$ and $D$ be (co)chain complexes, then there is a natural isomorphism (a linear graded map of degree $0$): + $$ H(C) \tensor H(D) \tot{\iso} H(C \tensor D), $$ + where we understand both tensors as graded. +\end{theorem} + \TODO{Discuss: -\titem The Künneth theorem (especially in the case of fields) -\titem The tensor algebra $T : Ch^\ast(\Q) \to \DGA_\Q$ and free cdga $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$ \titem Coalgebras and Hopf algebras? -\titem Define reduced/connected differential graded things \titem Singular (co)homology as a quick example? -} \ No newline at end of file +} diff --git a/thesis/notes/Minimal_Models.tex b/thesis/notes/Minimal_Models.tex new file mode 100644 index 0000000..e3778ca --- /dev/null +++ b/thesis/notes/Minimal_Models.tex @@ -0,0 +1,113 @@ + +\section{Minimal models} + +In this section we will discuss the so called minimal models. These are cdga's with the property that a quasi isomorphism between them is an actual isomorphism. + +\begin{definition} + An cdga $(A, d)$ is a \emph{Sullivan algebra} if + \begin{itemize} + \item $(A, d)$ is quasi-free (or semi-free), i.e. $A = \Lambda V$ is free as a cdga, and + \item $V$ has a filtration $V(0) \subset V(1) \subset \cdots \subset \bigcup{k \in \N} V(k) = V$ such that $d(V(k)) \subset \Lambda V(k-1)$. + \end{itemize} + + An cdga $(A, d)$ is a \emph{minimal (Sullivan) algebra} if in addition + \begin{itemize} + \item $d$ is decomposable, i.e. $\im(d) \subset \Lambda^{\geq 2}V$. + \end{itemize} +\end{definition} + +\begin{definition} + Let $(A, d)$ be any cdga. A \emph{(minimal) Sullivan model} is a (minimal) Sullivan algebra $(M, d)$ with a weak equivalence: + $$ (M, d) \we (A, d). $$ +\end{definition} + +The requirement that there exists a filtration can be replaced by a stronger statement. + +\begin{lemma} + Let $(A, d)$ be a cdga which is $1$-reduced, quasi-free and with a decomposable differential. Then $(A, d)$ is a minimal algebra. +\end{lemma} +\begin{proof} + Take $V(n) = \bigoplus_{k=0}^n V^k$ (note that $V^0 = v^1 = 0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v) = x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees: + $$ \deg{x} + \deg{y} = \deg{xy} = \deg{dv} = \deg{v} + 1 = n + 1. $$ + As $A$ is $1$-reduced we have $\deg{x}, \deg{y} \geq 2$ and so by the above $\deg{x}, \deg{y} \leq n-1$. Conclude that $d(V(k)) \subset \Lambda(V(n-1))$. +\end{proof} + + +\subsection{Existence} + +\begin{theorem} + Let $(A, d)$ be an $1$-connected cdga, then it has a minimal model. +\end{theorem} +\begin{proof} + We will construct a sequence of models $m_k: (M(k), d) \to (A, d)$ inductively. + \begin{itemize} + \item First define $V(0) = V(1) = 0$ and $m_0 = m_1 = 0$. Then set $V(2) = H^2(A)$ and define a map $m_2: V(2) \to A$ by picking representatives. + \item Suppose $m_k: (\Lambda V(k), d) \to (A, d)$ is constructed. Choose cocycles $a_\alpha \in A^{k+1}$ and $z_\beta \in (\Lambda V(k))^{k+2}$ such that $H^{k+1}(A) = \im(H^{k+1}(m_k)) \oplus \bigoplus_\alpha \k \cdot [a_\alpha]$ (so $m_k$ together with $a_\alpha$ span $H^{k+1}(A)$) and $\ker(H^{k+2}(m_k)) = \bigoplus_\beta \k \cdot [z_\beta]$. Note that $m_k z_\beta = db_\beta$ for some $b_\beta \in A$. + + Define $V(k+1) = \bigoplus_\alpha \k \cdot v'_\alpha \oplus \bigoplus \k \cdot v''_\beta$ and set $dv'_\alpha = 0$, $dv''_\beta = z_\beta$, $m_k(v'_\alpha) = a_\alpha$ and $m_k(v''_\beta) = b_\beta$. + \end{itemize} + This ends the construction. We will prove the following assertion for $k \geq 2$: + $$ H^i(m_k) \text{ is } \begin{cases} + \text{an isomorphism} &\text{ if } i \leq k \\ + \text{injective} &\text{ if } i = k + 1 + \end{cases}. $$ + \TODO{Finish proof: $m_k$ well behaved, above assertion.} +\end{proof} + + +\subsection{Uniqueness} + +Before we state the uniqueness theorem we need some more properties of minimal models. + +\begin{lemma} + Sullivan algebras are cofibrant. +\end{lemma} +\begin{proof} + Consider the following lifting problem, where $\Lambda V$ is a Sullivan algebra. + + \cimage[scale=0.5]{Sullivan_Lifting} + + By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ sucht that $p \circ \phi = g$. We will do this by induction. + \begin{itemize} + \item Suppose $\{v_\alpha\}$ is a basis for $V(0)$. Define $V(0) \to X$ by choosing preimages $x_\alpha$ such that $p(x_\alpha) = g(v_\alpha)$ ($p$ is surjective). Define $\phi(v_\alpha) = x_\alpha$. + \item Suppose $\phi$ has been defined on $V(n)$. Write $V(n+1) = V(n) \oplus V'$ and let $\{v_\alpha\}$ be a basis for $V'$. Then $dv_\alpha \in \Lambda V(n)$, hence $\phi(dv_\alpha)$ is defined and + $$ d \phi d v_\alpha = \phi d^2 v_\alpha = 0 $$ + $$ p \phi d v_\alpha = g d v_\alpha = d g v_\alpha. $$ + Now $\phi d v_\alpha$ is a cycle and $p \phi d v_\alpha$ is a boundary of $g v_\alpha$. By the following lemma there is a $x_\alpha \in X$ such that $d x_\alpha = \phi d v_\alpha$ and $p x_\alpha = g v_\alpha$. The former property proves that $\phi$ is a chain map, the latter proves the needed commutativity $p \circ \phi = g$. + \end{itemize} +\end{proof} + +\begin{lemma} + Let $p: X \to Y$ be a trivial fibration, $x \in X$ a cycle, $p(x) \in Y$ a boundary of $y' \in Y$. Then there is a $x' \in X$ such that + $$ dx' = x \quad\text{ and }\quad px' = y'. $$ +\end{lemma} +\begin{proof} + We have $p^\ast [x] = [px] = 0$, since $p^\ast$ is injective we have $x = d \overline{x}$ for some $\overline{x} \in X$. Now $p \overline{x} = y' + db$ for some $b \in Y$. Choose $a \in X$ with $p a = b$, then define $x' = \overline{x} - da$. Now check the requirements: $p x' = p \overline{x} - p a = y'$ and $d x' = d \overline{x} - d d a = d \overline{x} = x$. +\end{proof} + +\begin{lemma} + Let $f: X \we Y$ be a weak equivalence between cdga's and $M$ a minimal model for $X$. Then $f$ induces an bijection: + $$ f_\ast: [M, X] \tot{\iso} [M, Y]. $$ +\end{lemma} +\begin{proof} + If $f$ is surjective this follows from the fact that $M$ is cofibrant and $f$ being a trivial fibration (see \cite[lemma 4.9]{dwyer}). + + In general we can construct a cdga $Z$ and trivial fibrations $X \to Z$ and $Y \to Z$ inducing bijections: + $$ [M, X] \tot{\iso} [M, Z] \toti{\iso} [M, Y], $$ + compatible with $f_\ast$. \cite[Proposition 12.9]{felix}. +\end{proof} + +\begin{lemma} + Let $\phi: (M, d) \we (M', d')$ be a weak equivalence between minimal algebras. Then $\phi$ is an isomorphism. +\end{lemma} +\begin{proof} + Let $M$ and $M'$ be generated by $V$ and $V'$. Then $\phi$ induces a weak equivalence on the linear part $\phi_0: V \we V'$ \cite[Theorem 1.5.10]{loday}. Since the differentials are decomposable, their linear part vanishes. So we see that $\phi_0: (V, 0) \tot{\iso} (V', 0)$ is an isomorphism. + Conclude that $\phi = \Lambda \phi_0$ is an isomorphism. +\end{proof} + +\begin{theorem} + Let $m: (M, d) \we (A, d)$ and $m': (M', d') \we (A, d)$ be two minimal models. Then there is an isomorphism $\phi (M, d) \tot{\iso} (M', d')$ such that $m' \circ \phi \eq m$. +\end{theorem} +\begin{proof} + By the previous lemmas we have $[M', M] \iso [M', A]$. By going from right to elft we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism. +\end{proof} diff --git a/thesis/preamble.tex b/thesis/preamble.tex index 7d2e7a1..d7f98fd 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -78,6 +78,7 @@ \newcommand{\iso}{\cong} % isomorphic \newcommand{\eq}{\sim} % homotopic \newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name +\newcommand{\toti}[1]{\xleftarrow{\,\,{#1}\,\,}} % left arrow with name \newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name \DeclareMathOperator*{\im}{im} \DeclareMathOperator*{\colim}{colim} diff --git a/thesis/thesis.tex b/thesis/thesis.tex index 4bea684..a2ee497 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -19,14 +19,17 @@ Some general notation: \todo{leave this out, or define somewhere else?} \vspace{1cm} -\input{notes/Algebra} \vspace{2cm} -\input{notes/Free_CDGA} \vspace{2cm} -\input{notes/CDGA_Basic_Examples} \vspace{2cm} -\input{notes/Model_Categories} \vspace{2cm} -\input{notes/Model_Of_CDGA} \vspace{2cm} -\input{notes/CDGA_Of_Polynomials} \vspace{2cm} -\input{notes/Polynomial_Forms} \vspace{2cm} -\input{notes/A_K_Quillen_Pair} \vspace{2cm} +\newcommand{\myinput}[1]{\input{#1} \vspace{2cm}} + +\myinput{notes/Algebra} +\myinput{notes/Free_CDGA} +\myinput{notes/CDGA_Basic_Examples} +\myinput{notes/Model_Categories} +\myinput{notes/Model_Of_CDGA} +\myinput{notes/CDGA_Of_Polynomials} +\myinput{notes/Polynomial_Forms} +\myinput{notes/A_K_Quillen_Pair} +\myinput{notes/Minimal_Models} % \listoftodos