Fixes some todos

thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex

@@ -22,7 +22,7 @@ We furthermore have the following categorical properties of cdga's:
 
 In this chapter the ring $\k$ is assumed to be a field of characteristic zero. In particular the modules are vector spaces.
 
-\todo{augmentations?}
+\todo{augmentations?}\todo{Kunneth from appendix}
 
 \section{Cochain models for the $n$-disk and $n$-sphere}
 \input{notes/CDGA_Basic_Examples}

thesis/chapters/Introduction.tex

@@ -36,7 +36,7 @@ We assume the reader is familiar with category theory, basics from algebraic top
 
 We will fix the following notations and categories.
 \begin{itemize}
-	\item $\k$ will denote a field of characteristic zero. Modules, tensor products,\dots are understood as $\k$-modules, tensor products over $\k$,\dots. \todo{$\k$ doesn't always seem to work...}
+	\item $\k$ will denote a field of characteristic zero. Modules, tensor products,\dots are understood as $\k$-modules, tensor products over $\k$,\dots.
 	\item $\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out.
 	\item $\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories.
 	\item $\Ab$: category of abelian groups and group homomorphisms.

thesis/notes/Algebra.tex

@@ -34,7 +34,6 @@ Recall that the tensor product of modules distributes over direct sums. This def
 	The tensor product extends to graded maps. Let $f: A \to B$ and $g:X \to Y$ be two graded maps, then their tensor product $f \tensor g: A \tensor B \to X \tensor Y$ is defined as:
 	$$ (f \tensor g)(a \tensor x) = (-1)^{\deg{a}\deg{g}} \cdot f(a) \tensor g(x). $$
 \end{definition}
-\todo{graded tor}
 
 The sign is due to \emph{Koszul's sign convention}: whenever two elements next to each other are swapped (in this case $g$ and $a$) a minus sign appears if both elements are of odd degree. More formally we can define a swap map
 $$ \tau : A \tensor B \to B \tensor A : a \tensor b \mapsto (-1)^{\deg{a}\deg{b}} b \tensor a. $$

thesis/notes/Basics.tex

@@ -71,7 +71,7 @@ The following two theorems can be found in textbooks about homological algebra s
 	\[ \footnotesize \xymatrix @C=0.3cm{
 		0 \ar[r] & H(X; A) \tensor H(Y; A) \ar[r] & H(X \times Y; A) \ar[r] & \Tor_{\ast-1}(H(X; A), H(Y; A)) \ar[r] & 0
 	},\]
-	where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded.
+	where the $H(X; A)$, $H(X; A)$ and their tensor product are considered as graded modules. The Tor group is graded as $\Tor_n(A, B) = \bigoplus_{i+j=n}(A_i, B_j)$.
 }
 
 \section{Consequences for rational homotopy theory}

thesis/notes/CDGA_Basic_Examples.tex

@@ -25,7 +25,7 @@ Those cocycles are in fact coboundaries (using that $\k$ is a field of character
 $$ c b^k = \frac{1}{k} d(b^{k+1}) $$
 $$ c^k = d(b c^{k-1}) $$
 
-There are no additional cocycles in $\Lambda D(n)$ besides the constants and $c$. So we conclude that $\Lambda D(n)$ is acyclic as an \todo{augmented?} algebra. In other words $\Lambda(j_n): \k \to \Lambda D(n)$ is a quasi isomorphism.
+There are no additional cocycles in $\Lambda D(n)$ besides the constants and $c$. So we conclude that $\Lambda D(n)$ is acyclic as an augmented algebra. In other words $\Lambda(j_n): \k \to \Lambda D(n)$ is a quasi isomorphism.
 
 The situation for $\Lambda S(n)$ is easier as it has only one generator (as algebra). For even $n$ this means it is given by polynomials in $a$. For odd $n$ it is an exterior algebra, meaning $a^2 = 0$. Again the sets $\Lambda(I) = \{ \Lambda(i_n) : \Lambda S(n+1) \to \Lambda D(n) \I n \in \N\}$ and $\Lambda(J) = \{ \Lambda(j_n) : \k \to \Lambda D(n) \I n \in \N\}$ play an important role.
 

thesis/notes/Rationalization.tex

@@ -36,7 +36,7 @@ $$ H_i(S^n_\Q) = \begin{cases}
 	0,  &\text{ otherwise.}
 \end{cases} $$
 
-By the Serre-Hurewicz theorem (\TheoremRef{serre-hurewicz}, with $\C$ the class of uniquely divisible groups) we see that $S^n_\Q$ is indeed rational. Then by the Serre-Whitehead theorem (\TheoremRef{serre-whitehead}, with $\C$ the class of torsion groups) the inclusion map $S^n \to S^n_\Q$ is a rationalization.
+By the Serre-Hurewicz theorem (\CorollaryRef{rational-hurewicz-2}) we see that $S^n_\Q$ is indeed rational. Then by the Serre-Whitehead theorem (\CorollaryRef{rational-whitehead}) the inclusion map $S^n \to S^n_\Q$ is a rationalization.
 
 \Corollary{rationalization-Sn}{
 	The inclusion $S^n \to S^n_\Q$ is a rationalization.

thesis/notes/Serre.tex

@@ -23,8 +23,8 @@ Serre gave weaker axioms for his classes and proves some of the following lemmas
 		\item The class $\C$ of all uniquely divisible groups. Note that these groups can be given a unique $\Q$-vector space structure (and conversely every $\Q$-vector space is uniquely divisible). 
 	\end{itemize}
 }
-\todo{refer to Moerdijk? for $H(\Z_p) =$ torsion.}
+
-As noted by Hilton in \cite{hilton} we think of Serre classes as a generalized 0. This means that we can also express some kind of generalized injective and surjectivity. Here we only need the notion of a $\C$-isomorphism:
+The first three axioms of Serre classes are easily checked. For the group homology we find a calculation of the group homology of cyclic groups in \cite{moerdijk}. The group homology itself is also a torsion group, this result extends to all torsion groups. As noted by Hilton in \cite{hilton} we think of Serre classes as a generalized 0. This means that we can also express some kind of generalized injective and surjectivity. Here we only need the notion of a $\C$-isomorphism:
 
 \Definition{serre-class-maps}{
 	Let $\C$ be a Serre class and let $f: A \to B$ be a map of abelian groups. Then $f$ is a \Def{$\C$-isomorphism} if both the kernel and cokernel lie in $\C$.
@@ -80,7 +80,7 @@ In the following arguments we will consider fibrations and need to compute homol
 	Let $\C$ be a Serre class and $G \in \C$. Then for all $n > 0$ and all $i > 0$ we have $H_i(K(G, n)) \in \C$.
 }
 \Proof{
-	We prove this by induction on $n$. The base case $n = 1$ follows from group homology as the construction \todo{which?} of $K(G, 1)$ can be used to obtain a projective resolution of $\Z$ as $\Z[G]$-module \todo{reference}. This then identifies the homology of the Eilenberg-MacLane space with the group homology which is in $\C$ by the axioms:
+	We prove this by induction on $n$. The base case $n = 1$ follows from group homology. By considering the nerve of $G$ we can construct $K(G,1)$. This construction can be related to the bar construction as found in \cite{moerdijk}. This then identifies the homology of the Eilenberg-MacLane space with the group homology which is in $\C$ by the axioms:
 	$$ H_i(K(G, 1); \Z) \iso H_i(G; \Z) \in \C. $$
 
 	Suppose we have proven the statement for $n$. If we consider the case of $n+1$ we can use the path fibration to relate it to the case of $n$:
@@ -189,13 +189,18 @@ For the main theorem we need the following construction. \todo{referentie}
 }
 
 Combining this lemma and \TheoremRef{serre-hurewicz} we get the following corollary for rational homotopy theory:
-
 \Corollary{rational-hurewicz}{
 	(Rational Hurewicz Theorem)
 	Let $X$ be a $1$-connected space. If $\pi_i(X) \tensor \Q = 0$ for all $i < n$, then $H_i(X; \Q) = 0$ for all $i < n$. Furthermore we have an isomorphism for all $i \leq n$:
 	$$ \pi_i(X) \tensor \Q \tot{\iso} H_i(X; \Q) $$
 }
 
+By using the class of $\Q$-vector spaces we get a dual theorem.
+\Corollary{rational-hurewicz-2}{
+	(Rational Hurewicz Theorem 2)
+	Let $X$ be a $1$-connected space. The homotopy groups $\pi_i(X)$ are $\Q$-vector spaces for all $i > 0$ if and only if $H_i(X)$ are $\Q$-vector spaces for all $i > 0$.
+}
+
 \todo{$\pi$ is $\Q$ local iff $H$ is}
 
 \TheoremRef{serre-whitehead} also applies verbatim to rational homotopy theory. However we would like to avoid the assumption that $\pi_2(f)$ is surjective. In \cite{felix} we find a way to work around this.

thesis/references.bib

@@ -127,6 +127,14 @@
 	publisher={Cambridge University Press}
 }
 
+@misc{moerdijk,
+	title={Notes on Homological Algebra},
+	author={Moerdijk, I.},
+	year={2008},
+	institution = {University of Utrecht},
+	howpublished = {University Lecture}
+}
+
 @article{neisendorfer,
 	title={Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces},
 	author={Neisendorfer, J.},