Adds more on the surjectivity trick. Updates todos

thesis/chapters/CDGA_As_Algebraic_Model_For_Rational_Homotopy_Theory.tex

@@ -22,6 +22,8 @@ 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?}
+
 \section{Cochain models for the $n$-disk and $n$-sphere}
 \input{notes/CDGA_Basic_Examples}
 

thesis/notes/A_K_Quillen_Pair.tex

@@ -134,7 +134,7 @@ Now the Tor group appearing in the theorem can be computed via a \emph{bar const
 Another exposition of this corollary can be found in \cite[Section 8.4]{berglund}. A very brief summary of the above statement is that $A$ sends homotopy pullbacks to homotopy pushout (assuming some connectedness).
 
 \section{Equivalence on rational spaces}
-For the equivalence of rational spaces and cdga's we need that the unit and counit of the adjunction in \CorollaryRef{minimal-model-adjunction} are in fact weak equivalences for rational spaces. More formally: for any (automatically cofibrant) $X \in \sSet$ and any minimal model $A \in \CDGA$, both rational, $1$-connected and of finite type, the following two natural maps are weak equivalences:
+For the equivalence of rational spaces and cdga's we need that the unit and counit of the adjunction in \CorollaryRef{minimal-model-adjunction} are in fact weak equivalences for rational spaces. More formally: for any (automatically cofibrant) $X \in \sSet$ and any minimal model $A \in \CDGA$, both rational, $1$-connected and of finite type \todo{undefined!}, the following two natural maps are weak equivalences:
 \begin{align*}
 	X &\to K(M(X)) \\
 	A &\to M(K(A))
@@ -198,7 +198,7 @@ In particular if the vector space $V'$ is finitely generated, we can repeat this
 	Now $V(n)$ is finitely generated for all $n$ by assumption. By the inductive procedure above we see that $(\Lambda V(n), d) \to A(K(\Lambda V(n), d))$ is a weak equivalence for all $n$. Hence $(\Lambda V, d) \to A(K(\Lambda V, d))$ is a weak equivalence.
 }
 
-Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by \CorollaryRef{serre-whitehead} (the Serre-Whitehead theorem). To be precise: for a simply connected rational space $X$ the map $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X))) \to A(X)$ is a weak equivalence.
+\todo{finite type!}Now we want to prove that $X \to K(M(X))$ is a weak equivalence for a simply connected rational space $X$ of finite type. For this, we will use that $A$ preserves and detects such weak equivalences by \CorollaryRef{serre-whitehead} (the Serre-Whitehead theorem). To be precise: for a simply connected rational space $X$ the map $X \to K(M(X))$ is a weak equivalence if and only if $A(K(M(X))) \to A(X)$ is a weak equivalence.
 
 \Lemma{}{
 	The map $X \to K(M(X))$ is a weak equivalence for simply connected rational spaces $X$ of finite type.

thesis/notes/Algebra.tex

@@ -34,6 +34,7 @@ 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
 $$ \tau : A \tensor B \to B \tensor A : a \tensor b \mapsto (-1)^{\deg{a}\deg{b}} b \tensor a. $$
@@ -83,6 +84,7 @@ It is not hard to see that the definition of a dga precisely defines the monoida
 Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i = 0$ for $i < 0$. Similarly if $M^i = 0$ for all $i < 0$, then $M$ is a \emph{cochain algebra}.
 
 \todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary cdga's.}
+\todo{Augmentations}
 
 
 \Remark{orthogonal-definition}{
@@ -108,6 +110,7 @@ If the module has more structure as discussed above, homology will preserve this
 		\item If $M$ has an algebra structure, then so does $H(M)$, given by
 		\[ [z_1] \cdot [z_2] = [z_1 \cdot z_2] \]
 		\item If $M$ is a commutative algebra, so is $H(M)$.
+		\todo{augmented}
 	\end{itemize}
 }
 Of course the converses need not be true. For example the singular cochain complex associated to a space is a graded differential algebra which is \emph{not} commutative. However, by taking homology one gets a commutative algebra.
@@ -115,9 +118,9 @@ Of course the converses need not be true. For example the singular cochain compl
 Note that taking homology of a differential graded module (or algebra) is functorial. Whenever a map $f: M \to N$ of differential graded modules (or algebras) induces an isomorphism on homology, we say that $f$ is a \emph{quasi isomorphism}.
 
 \begin{definition}
-	Let $M$ be a graded module. We say that $M$ is $n$-reduced if $M_i = 0$ for all $i \leq n$. Similarly we say that a graded algebra $A$ is $n$-reduced if $A_i = 0$ for all $1 \leq i \leq n$ and $\eta: \k \tot{\iso} A_0$.
+	Let $M$ be a graded module. We say that $M$ is $n$-reduced if $M_i = 0$ for all $i \leq n$. Similarly we say that a graded \todo{augmented} algebra $A$ is $n$-reduced if $A_i = 0$ for all $1 \leq i \leq n$ and $\eta: \k \tot{\iso} A_0$.
 
-	Let $(M, d)$ be a chain complex (or algebra). We say that $M$ is $n$-connected if $H(M)$ is $n$-reduced as graded module (resp. algebra). Similarly for cochain complexes.
+	Let $(M, d)$ be a chain complex (or algebra). We say that $M$ is $n$-connected if $H(M)$ is $n$-reduced as graded module (resp. \todo{augmented} algebra). Similarly for cochain complexes.
 \end{definition}
 
 

thesis/notes/Basics.tex

@@ -70,7 +70,7 @@ 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. \todo{Geef algebraische versie voor ketencomplexen? en cohomology?}
+	where $H(X; A)$ and $H(X; A)$ are considered as graded modules and their tensor product and torsion groups are graded.
 }
 
 \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 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 \todo{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/CDGA_Of_Polynomials.tex

@@ -53,7 +53,7 @@ One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace
 }
 
 \Lemma{apl-kan-complex}{
-	$\Apl^k$ is a Kan complex.
+	$\Apl^k$ is a Kan complex. \todo{also without $k$?}
 }
 \Proof{
 	By the simple fact that $\Apl^k$ is a simplicial group, it is a Kan complex \cite{goerss}.

thesis/notes/Free_CDGA.tex

@@ -49,3 +49,4 @@ Again this extends to differential graded modules (i.e. the ideal is preserved b
 \end{lemma}
 
 We can now easily construct cdga's by specifying generators and their differentials.
+\todo{augmented}

thesis/notes/Homotopy_Augmented_CDGA.tex

@@ -46,7 +46,7 @@ The second observation is that $Q$ is nicely behaved on tensor products and coke
 	\[ Q(\coker(f)) \iso \coker(Qf). \]
 }
 \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:
+	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 $\overline{B} / f(\overline{A})$ \todo{$B / f(\overline{A})B$}. 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})}

thesis/notes/Homotopy_Relations_CDGA.tex

@@ -31,7 +31,7 @@ Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{rig
 	$$ [A, X] = \Hom_{\CDGA_\k}(A, X) / \simeq. $$
 }
 
-The results from model categories immediately imply the following results. \todo{Refereer expliciet}
+The results from model categories immediately imply the following results. Here we use Lemma \ref{lem:left_homotopy_properties}, \ref{lem:right_homotopy_properties} and \ref{lem:weak_strong_homotopy}.
 \Corollary{cdga_homotopy_properties}{
 	Let $A$ be cofibrant.
 	\begin{itemize}
@@ -42,9 +42,15 @@ The results from model categories immediately imply the following results. \todo
 			f_\ast: [Z, A] &\tot{\iso} [Z, X], \\
 			f^\ast: [X, Z] &\tot{\iso} [A, X].
 		\end{align*}
-		\todo{De eerste werkt ook als $i$ gewoon een w.e. is. (Gebruik factorizatie.)}
 	\end{itemize}
 }
+\Remark{cdga-weak-eq-bijection}{
+	By \RemarkRef{cdga-mc5a-left-inverse} we can generalize the second item to arbitrary weak equivalences: If $A$ is cofibrant and $f : X \to Y$ a weak equivalence, then the induced map $f_\ast : [A, X] \to [A, Y]$ is a bijection, as seen from the following diagram:
+	\[ \xymatrix{
+		& [A, X \tensor C] \ar[dl]_{\overline{\phi}_\ast}^\iso \ar[dr]^{\psi_\ast}_\iso & \\
+		[A, X] \ar[rr]^{f_\ast} & & [A, Y]
+	}\]
+}
 
 \Lemma{cdga-homotopy-homology}{
 	Let $f, g: A \to X$ be two homotopic maps, then $H(f) = H(g): HA \to HX$.

thesis/notes/Minimal_Models.tex

@@ -65,7 +65,7 @@ It is clear that induction will be an important technique when proving things ab
 
 	Complete the construction by taking the union: $V = \bigcup_k V(k)$. Clearly $H(m)$ is surjective, this was established in the first step. Now if $H(m)[z] = 0$, then we know $z \in \Lambda V(k)$ for some stage $k$ and hence by construction is was killed, i.e. $[z] = 0$. So we see that $m$ is a quasi isomorphism and by construction $(\Lambda V, d)$ is a Sullivan algebra.
 
-	\todo{Rewrite this section} Now assume that $(A, d)$ is $r$-connected ($r \geq 1$), this means that $H^i(A) = 0$ for all $1 \leq i \leq r$, and so $V(0)^i = 0$ for all $i \leq r$. Now $H(m_0)$ is injective on $\Lambda^{\leq 1} V(0)$, and so the defects are in $\Lambda^{\geq 2} V(0)$ and have at least degree $2(r+1)$. This means two things in the first inductive step of the construction. First, the newly added elements have decomposable differential. Secondly, these elements are at least of degree $2(r+1) - 1$. After adding these elements, the new defects are in $\Lambda^{\geq 2} V(1)$ and have at least degree $2(2(r+1) - 1)$. We see that as the construction continues, the degrees of adjoined elements go up. Hence $V^i = 0$ for all $i \leq r$ and by \LemmaRef{1-reduced-minimal-model} $(\Lambda V, d)$ is minimal.
+	\todo{Rewrite this section} Now assume that $(A, d)$ is $r$-connected ($r \geq 1$), this means that $H^i(A) = 0$ for all $1 \leq i \leq r$, and so $V(0)^i = 0$ for all $i \leq r$. Now $H(m_0)$ is injective on $\Lambda^{\leq 1} V(0)$, and so the defects are in $\Lambda^{\geq 2} V(0)$ and have at least degree $2(r+1)$. This means two things in the first inductive step of the construction. First, the newly added elements have decomposable differential. Secondly, these elements are at least of degree $2(r+1) - 1$. After adding these elements, the new defects are in $\Lambda^{\geq 2} V(1)$ and have at least degree $2(2(r+1) - 1)$. We see that as the construction continues, the degrees of adjoined elements go up. Hence $V^i = 0$ for all $i \leq r$ and by \LemmaRef{1-reduced-minimal-model} \todo{does not apply} $(\Lambda V, d)$ is minimal.
 \end{proof}
 
 
@@ -103,17 +103,12 @@ Before we state the uniqueness theorem we need some more properties of minimal m
 	We have $p^\ast [x] = [px] = 0$, since $p^\ast$ is injective we have $x = d \overline{x}$ for some $\overline{x} \in X$. Now $p \overline{x} = y' + db$ for some $b \in Y$. Choose $a \in X$ with $p a = b$, then define $x' = \overline{x} - da$. Now check the requirements: $p x' = p \overline{x} - p a = y'$ and $d x' = d \overline{x} - d d a = d \overline{x} = x$.
 \end{proof}
 
-\Lemma{minimal-model-bijection}{
+As minimal models are cofibrant \RemarkRef{cdga-weak-eq-bijection} immediately implies the following.
+
+\Corollary{minimal-model-bijection}{
 	Let $f: X \we Y$ be a weak equivalence between cdga's and $M$ a minimal algebra. Then $f$ induces an bijection:
 	$$ f_\ast: [M, X] \tot{\iso} [M, Y]. $$
 }
-\begin{proof}
-	If $f$ is surjective this follows from the fact that $M$ is cofibrant and $f$ being a trivial fibration, see \CorollaryRef{cdga_homotopy_properties}.
-
-	By \RemarkRef{cdga-mc5a-left-inverse}, we can turn $f$ into two trivial fibrations (going in different directions). Hence we are in the above situation and we find bijections
-	$$ [M, X] \toti{\iso} [M, \Lambda C(Y)] \tot{\iso} [M, Y], $$
-	compatible with $f_\ast$.
-\end{proof}
 
 \begin{lemma}
 	Let $\phi: (M, d) \we (M', d')$ be a weak equivalence between minimal algebras. Then $\phi$ is an isomorphism.
@@ -131,7 +126,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m
 	Let $m: (M, d) \we (A, d)$ and $m': (M', d') \we (A, d)$ be two minimal models for $A$. Then there is an isomorphism $\phi (M, d) \tot{\iso} (M', d')$ such that $m' \circ \phi \eq m$.
 }
 \begin{proof}
-	By \LemmaRef{minimal-model-bijection} we have $[M', M] \iso [M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
+	By \CorollaryRef{minimal-model-bijection} we have $[M', M] \iso [M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism.
 \end{proof}
 
 The assignment to $X$ of its minimal model $M_X = (\Lambda V, d)$ can be extended to morphisms. Let $X$ and $Y$ be two cdga's and $f: X \to Y$ be a map. By considering their minimal models we get the following diagram.

thesis/notes/Model_Categories.tex

@@ -96,6 +96,7 @@ Of course the most important model category is the one of topological spaces. We
 		\item Cofibrations: all monomorphisms.
 	\end{itemize}
 }
+\todo{small object arg?}
 
 In this thesis we often restrict to $1$-connected spaces. The full subcategory $\Top_1$ of $1$-connected spaces satisfies MC2-MC5: the 2-out-of-3 property, retract property and lifting properties hold as we take the \emph{full} subcategory, factorizations exist as the middle space is $1$-connected as well. However $\Top_1$ does not have all limits and colimits.
 

thesis/notes/Polynomial_Forms.tex

@@ -56,11 +56,11 @@ In this section we will prove that the singular cochain complex is quasi isomorp
 For any $v \in \Apl_n^n$, we can write $v$ 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 geometrically on the $n$-simplex:
 $$ \int_n v = \int_0^1 \int_0^{1-x_n} \dots \int_0^{1 - x_2 - \dots - x_n} p(x_1, \dots, x_n) dx_1 dx_2 \dots dx_n, $$
 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):
-$$ \int_n x_1^{k_1} \dots x_n^{k_n} dx_1 \dots dx_n = \frac{k_1! \dots k_n!}{(k_1 + \dots + k_n + n)!}. $$
+$$ \int_n x_1^{k_1} \dots x_n^{k_n} dx_1 \dots dx_n = \frac{k_1! \dots k_n!}{(k_1 + \dots + k_n + n)!}. $$\todo{same as above}
 
-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
+\todo{rewrite?}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 $\oint_n: \Apl_n \to C_n$.
+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 $\oint_n(v) \in C_n$\todo{normalised}. 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 $\oint = \{\oint_n\}_n$ is a simplicial map and that each $\oint_n$ is a chain map, in other words $\oint : \Apl \to C$ is a simplicial chain map (of complexes). Let $\sigma: \Delta[n] \to \Delta[k]$, and $\sigma^\ast: \Apl_k \to \Apl_n$ its induced map. We need to prove $\oint_n \circ \sigma^\ast = \sigma^\ast \circ \oint_k$. We show this as follows:
 \begin{align*}
@@ -95,7 +95,7 @@ We will now prove that the map $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphis
 	We can apply our two functors to it, and use the natural transformation $\oint$ to obtain the following cube:
 	\cdiagram{Apl_C_Quasi_Iso_Cube}
 
-	Note that $A(\Delta[n]) \we C^\ast(\Delta[n])$ by \CorollaryRef{apl-c-quasi-iso}, $A(X) \we C^\ast(X)$ by assumption and $A(\del \Delta[n]) \we C^\ast(\del \Delta[n])$ by induction. Secondly note that both $A$ and $C^\ast$ send injective maps to surjective maps, so we get fibrations on the right side of the diagram. Finally note that the front square and back square are pullbacks, by adjointness of $A$ and $C^\ast$. Apply the cube lemma (\LemmaRef{cube-lemma}) to conclude that also $A(X') \we C^\ast(X')$.
+	Note that $A(\Delta[n]) \we C^\ast(\Delta[n])$ by \CorollaryRef{apl-c-quasi-iso}, $A(X) \we C^\ast(X)$ by assumption and $A(\del \Delta[n]) \we C^\ast(\del \Delta[n])$ by induction. Secondly note that both $A$ and $C^\ast$ send injective maps to surjective maps\todo{see 7.0.2, what about $C$?}, so we get fibrations on the right side of the diagram. Finally note that the front square and back square are pullbacks, by adjointness of $A$ and $C^\ast$. Apply the cube lemma (\LemmaRef{cube-lemma}) to conclude that also $A(X') \we C^\ast(X')$.
 
 	This proves $A(X) \we C^\ast(X)$ for any simplicial set with finitely many non-degenerate simplices. We can extend this to simplicial sets of finite dimension by attaching many simplices at once. For this we observe that both $A$ and $C^\ast$ send coproducts to products and that cohomology commutes with products:
 	$$ H(A(\coprod_\alpha X_\alpha)) \iso H(\prod_\alpha A(X_\alpha)) \iso \prod_\alpha H(A(X_\alpha)), $$
@@ -139,3 +139,5 @@ We will now prove that the map $\oint: A(X) \to C^\ast(X)$ is a quasi isomorphis
 
 	So by the five lemma we can conclude that the middle morphism is an isomorphism as well, proving the isomorphism $H^n(A(X)) \tot{\iso} H^n(C^\ast(X))$ for all $n$. This proves the statement for all $X$.
 }
+
+\todo{algebra structure}

thesis/notes/Rationalization.tex

@@ -31,7 +31,7 @@ Moreover we note that the generator $1 \in \pi_n(S^n)$ is sent to $1 \in \pi_n(S
 	The inclusion $S^1 \to S^1_\Q$ is a rationalization.
 }
 
-For $n>1$ we can resort to homology, which also commutes with filtered colimits \cite[14.6]{may}. By connectedness we have $H_0(S^n_\Q) = \Z$ and for $i \neq 0, n$ we have $H_i(S^n) = 0$, so in these cases the homology of the colimit is also $\Z$ and resp. $0$. For $i = n$ we can use the same sequence as above (or use the Hurewicz theorem) to conclude:
+For $n>1$ we can resort to homology, which also commutes with filtered colimits \cite[14.6]{may}. By connectedness we have $H_0(S^n_\Q) = \Z$ and for $i \neq 0, n$ we have $H_i(S^n) = 0$, so in these cases the homology of the colimit is also $\Z$ and resp. $0$ \todo{leesbaarder}. For $i = n$ we can use the same sequence as above (or use the Hurewicz theorem) to conclude:
 
 $$ H_i(S^n_\Q) = \begin{cases}
 	\Z, &\text{ if } i = 0 \\

thesis/preamble.tex

@@ -182,6 +182,7 @@
 \newcommand{\TheoremRef}{\RefTemp{Theorem}{thm}}
 \newcommand{\LemmaRef}{\RefTemp{Lemma}{lem}}
 \newcommand{\CorollaryRef}{\RefTemp{Corollary}{cor}}
+\newcommand{\RemarkRef}{\RefTemp{Remark}{rmk}}
 
 \newcommand{\DefinitionRef}{\RefTemp{Definition}{def}}