Fixes some typos

thesis/notes/Basics.tex

@@ -8,7 +8,7 @@ In the following definition \emph{space} is to be understood as a topological sp
 
 \Definition{rational-space}{
 	A space $X$ is a \emph{rational space} if
-	$$ \pi_i(X) \text{ is a $\Q$-vectorspace } \quad\forall i > 0. $$
+	$$ \pi_i(X) \text{ is a $\Q$-vector space } \quad\forall i > 0. $$
 }
 
 \Definition{rational-homotopy-groups}{
@@ -26,7 +26,7 @@ Note that for a rational space $X$, the homotopy groups are isomorphic to the ra
 	A map $f: X \to X_0$ is a \emph{rationalization} if $X_0$ is rational and $f$ is a rational homotopy equivalence.
 }
 
-Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence iff $f$ is a weak equivalence.
+Note that a weak equivalence (and hence also a homotopy equivalence) is always a rational homotopy theory. Furthermore if $f: X \to Y$ is a map between rational spaces, then $f$ is a rational homotopy equivalence if and only if $f$ is a weak equivalence.
 
 We will later see that any space admits a rationalization. The theory of rational homotopy theory is then the study of the homotopy category $\Ho_\Q(\Top) \iso \Ho(\Top_\Q)$, which is on its own turn equivalent to $\Ho(\sSet_\Q) \iso \Ho_\Q(\sSet)$.
 
@@ -37,26 +37,26 @@ We will now recall known results from algebraic topology, without proof. One can
 \Theorem{relative-hurewicz}{
 	(Relative Hurewicz) For any inclusion of spaces $A \subset X$ and all $i > 0$, there is a natural map
 	$$ h_i : \pi_i(X, A) \to H_i(X, A). $$
-	If furhtermore $(X,A)$ is $n$-connected, then the map $h_i$ is an isomorphism for all $i \leq n + 1$
+	If furthermore $(X,A)$ is $n$-connected, then the map $h_i$ is an isomorphism for all $i \leq n + 1$
 }
 
 \Theorem{serre-les}{
 	(Long exact sequence) Let $f: X \to Y$ be a Serre fibration, then there is a long exact sequence:
 	$$ \cdots \tot{\del} \pi_i(F) \tot{i_\ast} \pi_i(X) \tot{f_\ast} \pi_i(Y) \tot{\del} \cdots \to \pi_0(Y) \to \ast, $$
-	where $F$ is the fibre of $f$.
+	where $F$ is the fiber of $f$.
 }
 
 Using an inductive argument and the previous two theorems, one can show the following theorem (as for example shown in \cite{griffiths}).
 \Theorem{whitehead-homology}{
 	(Whitehead) For any map $f: X \to Y$ we have
 	$$ \pi_i(f) \text{ is an isomorphism } \forall 0 < i < r \iff H_i(f) \text{ is an isomorphism } \forall 0 < i < r. $$
-	In particular we see that $f$ is a weak equivalence iff it induces an isomorphism on homology.
+	In particular we see that $f$ is a weak equivalence if and only if it induces an isomorphism on homology.
 }
 
-The following two theorems can be found in textbooks about homological algebra, such as [Weibel].
+The following two theorems can be found in textbooks about homological algebra, such as [\cite{weibel}].
 \Theorem{universal-coefficient}{
 	(Universal Coefficient Theorem)
-	For any space $X$ and abelian group $A$, there are natural short exact sequcenes
+	For any space $X$ and abelian group $A$, there are natural short exact sequences
 	$$ 0 \to H_n(X) \tensor A \to H_n(X; A) \to \Tor(H_{n-1}(X), A) \to 0, $$
 	$$ 0 \to \Ext(H_{n-1}(X), A) \to H^n(X; A) \to \Hom(H_n(X), A) \to 0. $$
 }

thesis/notes/CDGA_Of_Polynomials.tex

@@ -29,7 +29,7 @@ $$ 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 $0$.
 
-\Lemma{apl-contracible}{
+\Lemma{apl-contractible}{
 	$\Apl^k$ is contractible.
 }
 \Proof{
@@ -74,7 +74,7 @@ Besides the simplicial structure of $\Apl$, there is also the structure of a coc
 	$\Apl_n$ 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.
+	This is clear for $\Apl_0 = \k \cdot 1$. 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 $H(\Apl_n) \iso \bigtensor_{i=1}^n H \Lambda(x_i, dx_i) \iso \bigtensor_{i=1}^n H \Lambda D(0) \iso \k \cdot [1]$.
 

thesis/notes/Model_Categories.tex

@@ -54,7 +54,7 @@
 		\item we write $f: A \cof B$ when $f$ is a cofibration and
 		\item we write $f: A \we B$ when $f$ is a weak equivalence.
 	\end{itemize}
-	Furthermore a map which is a fibration and a weak equivalence is callend a \Def{trivial fibration}, similarly we have \Def{trivial cofibration}.
+	Furthermore a map which is a fibration and a weak equivalence is called a \Def{trivial fibration}, similarly we have \Def{trivial cofibration}.
 }
 
 \Definition{model-cats-fibrant-cofibrant}{
@@ -88,7 +88,7 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival
 \end{tikzpicture}
 \end{center}
 
-The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abreviate left lifting property with LLP and right lifting property with RLP. We will not prove these statements, but only expose them because we use them throughout this thesis. One can find proofs in \cite{dwyer, may}.
+The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abbreviate left lifting property with LLP and right lifting property with RLP. We will not prove these statements, but only expose them because we use them throughout this thesis. One can find proofs in \cite{dwyer, may}.
 
 \Lemma{model-cats-characterization}{
 	Let $\cat{C}$ be a model category.

thesis/notes/Polynomial_Forms.tex

@@ -127,6 +127,6 @@ We will now prove that the map $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphis
 
 	\cimage[scale=0.5]{Apl_C_Quasi_Iso_LES}
 
-	So by the five lemma we can conclude that the middel morphism is an isomorphism as well, proving $H^n(A(X)) \tot{\iso} H^n(C^\ast(X))$ for all $n$. This proves the statement for all $X$.
+	So by the five lemma we can conclude that the middle morphism is an isomorphism as well, proving $H^n(A(X)) \tot{\iso} H^n(C^\ast(X))$ for all $n$. This proves the statement for all $X$.
 }
 

thesis/notes/Serre.tex

@@ -11,7 +11,7 @@ In this section we will prove the Whitehead and Hurewicz theorems in a rational
 	such that:
 	\begin{itemize}
 		\item $K(\pi_n(X), n-1) \cof X(n+1) \fib X(n)$ is a fiber sequence,
-		\item  There is a space $X'_n$ weakly equivelent to $X(n)$ such that $X(n+1) \ cof X'_n \fib K(\pi_n(X), n)$ is a fiber sequence, and
+		\item  There is a space $X'_n$ weakly equivalent to $X(n)$ such that $X(n+1) \ cof X'_n \fib K(\pi_n(X), n)$ is a fiber sequence, and
 		\item  $X(n)$ is $(n-1)$-connected and $\pi_i(X(n)) \iso \pi_i(X)$ for all $i \geq n$.
 	\end{itemize}
 }
@@ -62,10 +62,10 @@ Note that the map $0 \to C$ is a $\C$-iso for any $C \in \C$. \todo{Add some stu
 In the following arguments we will consider fibrations and need to compute homology thereof. Unfortunately there is no long exact sequence for homology of a fibration, however the following lemma expresses something similar. It is usually proven with spectral sequences,  \cite[Ch. 2 Thm 1]{serre}. However in \cite{kreck} we find a more elementary proof using cellular homology.
 
 \Lemma{kreck}{
-	Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $1$-connected spaces and $F$ its fibre. If $\RH_i(F) \in \C$ for all $i < n$, then
+	Let $\C$ be a Serre class. Let $p: E \fib B$ be a fibration between $1$-connected spaces and $F$ its fiber. If $\RH_i(F) \in \C$ for all $i < n$, then
 	\begin{itemize}
 		\item $H_i(E, F) \to H_i(B, b_0)$ is a $\C$-iso for $i \leq n+1$ and
-		\item $H_i(E) \to H_i(B)$ is a $\C$-iso forall $i \leq n$.
+		\item $H_i(E) \to H_i(B)$ is a $\C$-iso for all $i \leq n$.
 	\end{itemize}
 }
 \Proof{
@@ -104,7 +104,7 @@ In the following arguments we will consider fibrations and need to compute homol
 	It remains to show that $h_n$ is a $\C$-iso. Use the Whitehead tower from \LemmaRef{whitehead-tower} to obtain $\cdots \fib X(3) \fib X(2) = X$. Note that each $X(j)$ is also $1$-connected and that $X(2) = X(1) = X$.
 
 	\Claim{}{For all $j < n$ and $i \leq n$ the induced map $H_i(X(j+1)) \to H_i(X(j))$ is a $\C$-iso.}
-	Note that $X(j+1) \fib X(j)$ is a fibration with $F = K(\pi_j(X), j-1)$ as its fibre. So by \LemmaRef{homology-em-space} we know $H_i(F) \in \C$ for all $i$. Apply \LemmaRef{kreck} to obtain a $\C$-iso $H_i(X(j+1)) \to H_i(X(j))$ for all $j < n$ and all $i > 0$. This proves the claim.
+	Note that $X(j+1) \fib X(j)$ is a fibration with $F = K(\pi_j(X), j-1)$ as its fiber. So by \LemmaRef{homology-em-space} we know $H_i(F) \in \C$ for all $i$. Apply \LemmaRef{kreck} to obtain a $\C$-iso $H_i(X(j+1)) \to H_i(X(j))$ for all $j < n$ and all $i > 0$. This proves the claim.
 
 	Considering this claim for all $j < n$ gives a chain of $\C$-isos $H_i(X(n)) \to H_i(X(n-1)) \to \cdot \to H_i(X(2)) = H_i(X)$ for all $i \leq n$. Consider the following diagram: