Adds long exact sequence for homotopy groups of cdga's

thesis/notes/Homotopy_Augmented_CDGA.tex

@@ -18,6 +18,58 @@ In the next section homotopy groups of augmented cdga's will be defined. In orde
 		Define the \Def{augmentation ideal} of $A$ as $\overline{A} = \ker \counit$. Define the \Def{cochain complex of indecomposables} of $A$ as $QA = \overline{A} / \overline{A} \cdot \overline{A}$.
 } 
 
-The first observation one should make is that $Q$ is a functor from algebras to modules (or differential algebras to differential modules) which is particularly nice for free algebras, as we have that $Q \Lambda V = V$ for any (differential) module $V$.
+The first observation one should make is that $Q$ is a functor from algebras to modules (or differential algebras to differential modules) which is particularly nice for free (differential) algebras, as we have that $Q \Lambda V = V$ for any (differential) module $V$.
 
-\todo{tensor}
+The second observation is that $Q$ is nicely behaved on tensor products and cokernels.
+\Lemma{Q-preserves-copord}{
+	Let $A$ and $B$ be two augmented cdga's, then there is a natural isomorphism
+	\[ Q(A \tensor B) \iso Q(A) \oplus Q(B). \]
+}
+\Proof{
+	First note that the augmentation ideal is expressed as
+	$\overline{A \tensor B} = \overline{A} \tensor B \>+\> A \tensor \overline{B}$
+	and the product is
+	$\overline{A \tensor B} \cdot \overline{A \tensor B} = \overline{A} \tensor \overline{B} \>+\> \overline{A}\cdot\overline{A} \tensor \k \>+\> \k \tensor \overline{B}$.
+	With this we can prove the statement
+	\begin{align*}
+		Q(A \tensor B)
+		&= \frac{\overline{A} \tensor B \>+\> A \tensor \overline{B}}
+			{\overline{A} \tensor \overline{B} \>+\> \overline{A}\cdot\overline{A} \tensor \k \>+\> \k \tensor \overline{B}} \\
+		&\iso \frac{\overline{A} \tensor \k \>\oplus\> \k \tensor \overline{B}}
+			{\overline{A}\cdot\overline{A} \tensor \k \>\oplus\> \k \tensor \overline{B}\cdot\overline{B}}
+		= Q(A) \,\oplus\, Q(B).
+	\end{align*}
+}
+
+\Lemma{Q-preserves-coeq}{
+	Let $f : A \to B$ be a map of augmented cdga's, then there is a natural isomorphism
+	\[ Q(\coker(f)) \iso \coker(Qf). \]
+}
+\Proof{
+	First note that the cokernel of $f$ in the category of augmented cdga's is $\coker(f) = B / f(\overline{A})$ and that its augmentation ideal is $\overline{B} / f(\overline{A})$. Just as above we make a simple calculation, where $p: \overline{B} \to Q(B)$ is the projection map:
+	\begin{align*}
+		Q(\coker(f))
+		&= \frac{\overline{B} / f(\overline{A})}
+			{\overline{B} / f(\overline{A}) \cdot \overline{B} / f(\overline{A})} \\
+		&\iso \frac{\overline{B} / \overline{B}\cdot\overline{B}}
+			{pf(\overline{A})}
+		= \frac{Q(B)}{Qf(Q(A))}.
+	\end{align*}
+}
+
+\Corollary{Q-preserves-pushouts}{
+	Combining the two lemmas above, we see that $Q$ (as functor from augmented cdga's to cochain complexes) preserves pushouts.
+}
+
+Furthermore we have the following lemma which is of homotopical interest.
+
+\Lemma{Q-preserves-cofibs}{
+	If $f: A \to B$ is a cofibration of augmented cdga's, then $Qf$ is injective in positive degrees.
+}
+\Proof{
+	First we define an augmented cdga $U(n)$ for each positive $n$ as $U(n) = D(n) \oplus \k$ with trivial multiplication and where the term $\k$ is used for the unit and augmentation. Notice that the map $U(n) \to \k$ is a trivial fibration. By the lifting property we see that the induced map
+	\[ \Hom_\AugCDGA(Y, U(n)) \tot{f^\ast} \Hom_\AugCDGA(X, U(n)) \]
+	is surjective for each positive $n$. Note that maps from $X$ to $U(n)$ will send products to zero and that it is fixed on the augmentation. So there is a natural isomorphism $\Hom_\AugCDGA(X, U(n)) \iso \Hom_\k(Q(X)^n, \k)$. Hence
+	\[ \Hom_\k(Q(Y)^n, \k) \tot{(Qf)^\ast} \Hom_\k(Q(X)^n, \k) \]
+	is surjective, and so $Qf$ itself is injective in positive $n$.
+}

thesis/notes/Homotopy_Groups_CDGA.tex

@@ -6,18 +6,18 @@ As the eventual goal is to compare the homotopy theory of spaces with the homoto
 	$$ \pi^i(A) = H^i(QA). $$
 }
 
-This construction is functorial and, as the following lemma shows, homotopy invariant.
+This construction is functorial (since both $Q$ and $H$ are) and, as the following lemma shows, homotopy invariant.
 
 \Lemma{cdga-homotopic-maps-equal-pin}{
-	Let $f: A \to B$ be a map of augmented cdga's. Then there is an functorial induced map on the homotopy groups. Moreover if $g: A \to B$ is homotopic to $f$, then the induced maps are equal:
-	$$ f_\ast = g_\ast : \pi_\ast(A) \to \pi_\ast(B). $$
+	Let $f: A \to X$ and $g: A \to X$ be a maps of augmented cdga's. If $f$ and $g$ are homotopic, then the induced maps are equal:
+	$$ f_\ast = g_\ast : \pi_\ast(A) \to \pi_\ast(X). $$
 }
 \Proof{
-	Let $\phi: A \to B$ be a map of algebras. Then clearly we get an induced map $\overline{A} \to \overline{B}$ as $\phi$ preserves the augmentation. By composition we get a map $\phi': \overline{A} \to Q(B)$ for which we have $\phi'(xy) = \phi'(x)\phi'(y) = 0$. So it induces a map $Q(\phi): Q(A) \to Q(B)$. By functoriality of taking homology we get $f_\ast : \pi^n(A) \to \pi^n(B)$. \todo{functoriality is redundant with previous section}
+	Let $h: A \to \Lambda(t, dt) \tensor X$ be a homotopy. We will, just as in \LemmaRef{cdga-homotopy-homology}, prove that the maps $HQ(d_0)$ and $HQ(d_1)$ are equal, then it follows that $HQ(f) = HQ(d_1 h) = HQ(d_0 h) = HQ(g)$.
 
-	Now if $f$ and $g$ are homotopic, then there is a homotopy $h: A \to \Lambda(t, dt) \tensor B$. By the Künneth theorem we have:
-	$$ {d_0}_\ast = {d_1}_\ast : H(\Lambda(t, dt) \tensor Q(B)) \to H(Q(B)). $$
-	This means that $f_\ast = {d_1}_\ast h_\ast = {d_0}_\ast h_\ast = g_\ast$. \todo{detail}
+	Using \LemmaRef{Q-preserves-copord} we can identify the induced maps $Q(d_i) : Q(\Lambda(t, dt) \tensor X) \to Q(X)$ with maps
+	\[ Q(d_i) : Q(\Lambda(t, dt)) \oplus Q(A) \to Q(A). \]
+	Now $Q(\Lambda(t, dt)) = D(0)$ and hence it is acyclic, so when passing to homology, this term vanishes. In other words both maps ${d_i}_\ast : H(D(0)) \oplus H(Q(A)) \to H(Q(A))$ are the identity maps on $H(Q(A))$.
 }
 
 Consider the augmented cdga $V(n) = S(n) \oplus \k$, with trivial multiplication and where the term $\k$ is used for the unit and augmentation. This augmented cdga can be thought of as a specific model of the sphere. In particular the homotopy groups can be expressed as follows.
@@ -35,4 +35,27 @@ Consider the augmented cdga $V(n) = S(n) \oplus \k$, with trivial multiplication
 
 From now on the dual of a vector space will be denoted as $V^\ast = \Hom_\k(V, \k)$. So the above lemma states that there is a bijection $[A, V(n)] \iso \pi^n(A)^\ast$.
 
-\todo{long exact sequence}
+In topology we know that a fibration induces a long exact sequence of homotopy groups. In the case of cdga's a similar long exact sequence for a cofibration will exist.
+
+\Lemma{long-exact-cdga-homotopy}{
+	Given a pushout square of augmented cdga's
+	\[ \xymatrix{
+		A \ar[d]^-f \arcof[r]^-g \xypo & C \ar[d]^-i \\
+		B \ar[r]^-j & P
+	} \]
+	where $g$ is a cofibration. There is a natural long exact sequence
+	\[ \pi^o(V) \tot{(f_\ast, g_\ast)} \pi^0(B) \oplus \pi^0(C) \tot{j_\ast - i_\ast} \pi^0(P) \tot{\del} \pi^1(A) \to \cdots \]
+}
+\Proof{
+	First note that $j$ is also a cofibration. By \LemmaRef{Q-preserves-cofibs} the maps $Qg$ and $Qj$ are injective in positive degrees. By applying $Q$ we get two exact sequence (in positive degrees) as shown in the following diagram. By the fact that $Q$ preserves pushouts (\LemmaRef{Q-preserves-pushouts}) the cokernels coincide.
+	\[ \xymatrix {
+		0 \ar[r] & Q(A) \ar[r] \ar[d] \xypo & Q(C) \ar[r] \ar[d] & \coker(f_\ast) \ar[r] \ar[d] & 0 \\
+		0 \ar[r] & Q(B) \ar[r] & Q(P) \ar[r] & \coker(f_\ast) \ar[r] & 0
+	} \]
+	Now the well known Mayer-Vietoris exact sequence can be constructed. This proves the statement.
+}
+
+\Corollary{long-exact-cdga-homotopy}{
+	When we take $B = \k$ in the above situation, we get a long exact sequence
+	\[ \pi^0(A) \tot{g_\ast} \pi^0(C) \to \pi^0(\coker(g)) \to \pi^1(A) \to \cdots \]
+}

thesis/notes/Homotopy_Relations_CDGA.tex

@@ -1,7 +1,7 @@
 
 Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
 
-Consider the free cdga on one generator $\Lambda(t, dt)$\todo{same as $\Lambda D(0)$}, where $\deg{t} = 0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows:
+Consider the free cdga on one generator $\Lambda(t, dt)$, where $\deg{t} = 0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. Notice that this cdga is isomorphic to $\Lambda(D(0))$ as defined in the previous section. We define two \emph{endpoint maps} as follows:
 $$ d_0, d_1 : \Lambda(t, dt) \to \k $$
 $$ d_0(t) = 1, \qquad d_1(t) = 0, $$
 this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t) = 0$ and $d_1(1-t) = 1$. These two functions extend to tensor products as $d_0, d_1: \Lambda(t, dt) \tensor X \to \k \tensor X \tot{\iso} X$.
@@ -46,7 +46,7 @@ The results from model categories immediately imply the following results. \todo
 	\end{itemize}
 }
 
-\Lemma{cdga_homotopy_homology}{
+\Lemma{cdga-homotopy-homology}{
 	Let $f, g: A \to X$ be two homotopic maps, then $H(f) = H(g): HA \to HX$.
 }
 \Proof{

thesis/preamble.tex

@@ -66,6 +66,7 @@
 \newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}}	% cochain complexes
 \DeclareRobustCommand{\DGA}{\cat{DGA}}			% cochain algebras
 \DeclareRobustCommand{\CDGA}{\cat{CDGA}}		% commutative cochain algebras
+\DeclareRobustCommand{\AugCDGA}{\cat{CDGA^\ast}}% augmentedcommutative cochain algebras
 
 \newcommand{\cof}{\hookrightarrow}		% cofibration
 \newcommand{\fib}{\twoheadrightarrow}	% fibration