Fix

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

6-differentiaforms.tex

@@ -962,23 +962,23 @@ \section{De Rham cohomology and Poincar\'e lemma}
 \begin{proof}
 The statement is clear\footnote{What is $\dim(M)$ in this case? What kinds of differential forms can we define?} if $M$ is equal to a point.
 The rest follows applying Corollary~\ref{cor:deRhamIso}.
-\end{proof}:
+\end{proof}
 
 \begin{exercise}
   Let $M$ be a smooth manifold. Define the following spaces
   \begin{align}
-    Z^k(M) &:= \ker(d:\Omega^k(M) \to \Omega^{k+1}(M)) = \{\mathrm{closed $k$-forms on $M$}\},\\
-    B^k(M) &:= \im(d:\Omega^{k-1}(M) \to \Omega^{k}(M)) = \{\mathrm{exact $k$-forms on $M$}\}.
+    Z^k(M) &:= \ker(d:\Omega^{k}(M) \to \Omega^{k+1}(M)) = \{ \mbox{closed $k$-forms on $M$} \},\\
+    B^k(M) &:= \mathrm{im}(d:\Omega^{k-1}(M) \to \Omega^{k}(M)) = \{ \mbox{exact $k$-forms on $M$} \}.
   \end{align}
-  Then
+    Then
   \begin{equation}
-    H^k_{dR}(M) = Z^k(M)/B^k(M).
+    H^k_{\mathrm{dR}}(M) = Z^k(M)/B^k(M).
   \end{equation}
-  Reason on the structure of those spaces to prove the following statements:
-  \begin{itemize}
-    \item $H^0_{dR}(M) = \R^C$ where $C$ denotes the number of connected components of $M$;
-    \item if $M$ is $n$-dimensional, then $H^k_{dR} = \{0\}$ for all $k > n$.
-  \end{itemize}
+  Reasoning on the structure of those spaces, prove the following statements:
+  \begin{enumerate}
+    \item $H^0_{\mathrm{dR}}(M) = \R^C$ where $C$ denotes the number of connected components of $M$;
+    \item if $M$ is $n$-dimensional, then $H^k_{\mathrm{dR}} = \{0\}$ for all $k > n$.
+  \end{enumerate}
 \end{exercise}
 
 \begin{exercise}