Adds a start of Apl = C

thesis/notes/CDGA_Of_Polynomials.tex

@@ -29,10 +29,10 @@ $$ s_i(x_j) = \begin{cases}
 
 One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k \in \simplicial{\Mod{\k}}$, this is indeed a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$.
 
-\begin{lemma}
+\Lemma{apl-contracible}{
 	$\Apl^k$ is contractible.
-\end{lemma}
-\begin{proof}
+}
+\Proof{
 	We will prove this by defining an extra degeneracy $s: \Apl_n \to \Apl_{n+1}$. Define for $i = 1, \ldots, n$:
 	\begin{align*}
 		s(1) &= (1-x_0)^2 \\
@@ -55,18 +55,29 @@ One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace
 	So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s = 0$.
 
 	So we have an extra degeneracy $s: \Apl^k \to \Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl \to \ast$ is a weak equivalence.
-\end{proof}
+}
 
-\begin{lemma}
+\Lemma{apl-kan-complex}{
 	$\Apl_n^k$ is a Kan complex.
-\end{lemma}
-\begin{proof}
+}
+\Proof{
 	By the simple fact that $\Apl_n^k$ is a simplicial group, it is a Kan complex \cite{goerss}.
-\end{proof}
+}
 
-\begin{corollary}
+Combining these two lemmas gives us the following.
+\Corollary{apl-extendable}{
 	$\Apl^k \to \ast$ is a trivial fibration in the standard model structure on $\sSet$.
-\end{corollary}
+}
+
+Besides the simplicial structure of $\Apl$, there is also the structure of a cochain complex.
+\Lemma{apl-acyclic}{
+	$\Apl_n$ is acyclic, i.e. $H(\Apl_n) = \k \cdot [1]$.
+}
+\Proof{
+	This is clear foor $\Apl_0 = \k \cdot 1$. For $\Apl_1$ we see that $\Apl_1 = \Lambda(x_1, dx_1) \iso \Lambda D(0)$, which we proved to be acyclic in the previous section.
 
+	For general $n$ we can identify $\Apl_n \iso \bigtensor_{i=1}^n \Lambda(x_i, dx_i)$, because $\Lambda$ is left adjoint and hence preserves coproducts. By the Künneth theorem \TheoremRef{kunneth} we conclude $H(\Apl_n) \iso \bigtensor_{i=1}^n H \Lambda(x_i, dx_i) \iso \bigtensor_{i=1}^n H \Lambda D(0) \iso \k \cdot [1]$.
 
+	So indeed $\Apl_n$ is acyclic for all $n$.
+}
 

thesis/notes/Polynomial_Forms.tex

@@ -45,10 +45,27 @@ where the addition, multiplication and differential are defined pointwise. Concl
 
 \subsection{The singular cochain complex}
 
-Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's:
+Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's \todo{Choose: normalized or not?}:
 $$ C_n = C^\ast(\Delta^n; \k). $$
 The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n \to \Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions
 
 \cimage[scale=0.5]{C_Extension}
 
-where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X) \we C^\ast(X)$ for every $X \in \sSet$. Furthermore this map preserves multiplication on the homology algebras.
+where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X) \we C^\ast(X)$ for every $X \in \sSet$. Furthermore this map preserves multiplication on the homology algebras.
+
+
+\subsection{Integration and Stokes' theorem for polynomial forms}
+
+In this section we will prove that the singular cochain complex is quasi isomorphic to the cochain complex of polynomial forms. In order to do so we will define an integration map $\int_n : \Apl_n^n \to \k$, which will induce a map $\oint_n : \Apl_n \to C_n$. For the simplices $\Delta[n]$ we already showed the cohomology agrees by the acyclicity of $\Apl_n = A(\Delta[n])$ (\LemmaRef{apl-acyclic}).
+
+Let $v \in \Apl_n^n$, then we can always write it as $v = p(x_1, \dots, x_n)dx_1 \dots dx_n$ where $p$ is a polynomial in $n$ variables. If $\Q \subset \k \subset \mathbb{C}$ we can integrate analytically
+$$ \int_n v = \int_0^1 \int_0^{1-x_n} \dots \int_0^{1 - x_2 - \dots - x_n} p(x_1, \dots, x_n) dx_1 dx_2 \dots dx_n, $$
+which defines a well-defined linear map $\int_n : \Apl_n^n \to \k$. For general fields of characteristic zero we can define it formally on the generators of $\Apl_n^n$ (as vector space):
+$$ \int_n x_1^{k_1} \dots x_n^{k_n} dx_1 \dots dx_n = \frac{k_1! \dots k_n!}{(k_1 + \dots + k_n)!}. $$
+
+Let $x$ be a $k$-simplex of $\Delta[n]$, i.e. $x: \Delta[k] \to \Delta[n]$. Then $x$ induces a linear map $x^\ast: \Apl_n \to \Apl_k$. Let $v \in \Apl_n^k$, then $x^\ast(v) \in \Apl_k^k$, which we can integrate. Now define
+$$ \oint_n(v)(x) = (-1)^\frac{k(k-1)}{2} \int_n x^\ast(v). $$
+Note that $\oint_n(v): \Delta[n] \to \k$ is just a map, we can extend this linearly to chains on $\Delta[n]$ to obtain $\oint_n(v): \Z\Delta[n] \to \k$, in other words $\oint_n(v) \in C_n$. By linearity of $\int_n$ and $x^\ast$, we have a linear map $\oint_n: \Apl_n \to C_n$.
+
+Next we will show that $\oint_n$ is a chain map and taking $\oint = \{\oint_n\}_n$ gives a simplicial chain map.
+