diff --git a/thesis/notes/Homotopy_Augmented_CDGA.tex b/thesis/notes/Homotopy_Augmented_CDGA.tex index 4f3fa31..4741a53 100644 --- a/thesis/notes/Homotopy_Augmented_CDGA.tex +++ b/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 $n$. +} diff --git a/thesis/notes/Homotopy_Groups_CDGA.tex b/thesis/notes/Homotopy_Groups_CDGA.tex index c8cba90..11d532f 100644 --- a/thesis/notes/Homotopy_Groups_CDGA.tex +++ b/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 + \[ \pi^0(A) \tot{g_\ast} \pi^0(C) \to \pi^0(\coker(g)) \to \pi^1(A) \to \cdots \] +} diff --git a/thesis/notes/Homotopy_Relations_CDGA.tex b/thesis/notes/Homotopy_Relations_CDGA.tex index 1517059..00c5413 100644 --- a/thesis/notes/Homotopy_Relations_CDGA.tex +++ b/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{ diff --git a/thesis/preamble.tex b/thesis/preamble.tex index 0aef805..a26f301 100644 --- a/thesis/preamble.tex +++ b/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