My LaTeX template

Today I tried going to a lecture and as expected still learned nothing. Seriously, how the heck do people learn at lectures? I mean, I can kind of understand how people learn during history or psychology lectures since those are heavy on story-telling, but there are no stories in real mathematics and so the college tradition of learning through oral instruction seems a bizarre oral-fixation, in grave need of therapy.

Anyway, I tried to at least try live-LaTeX-transcription of what's written on the board, and almost managed to do that, until I tried compiling and the thing does not compile since the tikz-cd package for drawing commutative diagrams doesn't work, and debugging spent so much time that I completely lost hope of keeping up, not that I originally had any hope of keeping up even without note-taking.

Because really, either it's too easy or it's too hard, it's never at the right speed. If it's too slow I get bored, and lose concentration. If it's too fast I can't understand and lose hope. Books never frustrate me like lectures.

Anyway, here's what I ended up using as a latex template. Might be useful for you.





\documentclass[a4paper, 12pt]{article}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{bbm}
\usepackage{color}
\usepackage{hyperref}
\usepackage{url}
\usepackage{graphicx}
\usepackage{enumerate}
\usepackage{wrapfig}
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{babel}
\usepackage [english]{babel}
\usepackage [autostyle, english = american]{csquotes}
\MakeOuterQuote{"}

\graphicspath{{img/}}

\theoremstyle{plain}% default
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{coro}{Corollary}

\theoremstyle{definition}
\newtheorem{defi}{Definition}[section]
\newtheorem{conj}{Conjecture}[section]
\newtheorem{exam}{Example}[section]

\theoremstyle{remark}
\newtheorem*{rem}{Remark}
\newtheorem*{note}{Note}
\newtheorem{case}{Case}

\title{Poincar\'e Duality}
\author{Twilight Sparkle}
\date{\vspace{-4ex}} % remove to generate date

\begin{document}

\maketitle

\begin{defi}
    A \textbf{manifold} of dimension $n$ is a space $M$ such that $\forall x\in M$ has a neighborhood $U$ homeomorphic to $\mathbb{R}^n$. $M$ is \textbf{closed} if and only if it is compact and without boundary.
\end{defi}
The local homology of $M$ is then 
\[H_i(M, M/\{x\} )\cong \begin{cases}
\mathbb{Z} \quad \text{if } i = n, \\
0 \quad \text{ else}
\end{cases}\]

\begin{defi}
A \textbf{local orientation} of $M$ at $x$ is a choice $\mu_x$ of $H_n(M, M/\{x\} )\cong \mathbb{Z}$ of generator ($\pm 1$).
\end{defi}
We can compare local orientation of "nearby" points. Suppose $x, y\in B \subset U \subset  M$ with $U \cong  \mathbb{R}^n$. Then $\mu_x, \mu_y$ are compatible if 

\[\begin{tikzcd}
{H_n(M, M/\{x\})} &  & {H_n(M, M/U)} \arrow[ll, "\cong"] &  & {H_n(M, M/\{y\})} \arrow[ll, "\cong"] \\
\mu_x &  & \mu_U \arrow[ll, maps to] &  & \mu_y \arrow[ll, maps to]
\end{tikzcd}\]

\begin{defi}\label{defi:orientation}
    An \textbf{orientation} of $M$ is a compatible family of local orientations $\mu_x$ for $\forall x\in M$. 
\end{defi}

\begin{exam}
    $\mathbb{R}P^3$ is \textbf{not} orientable.
\end{exam}

\begin{thm}\label{thm:poincare_duality}
    If $M$s a closed, oriented $n$-manifold, then there is an isomorphism 
\begin{tikzcd}
{H^k(M, R)} \arrow[rr, "\cong"] &  & {H_{n-k}(M, R)}
\end{tikzcd}
for any commutative ring $R$ and any $k\in \mathbb{Z}$.
\end{thm}

\begin{rem}
    For $R = \mathbb{Z}/2$, we don't need $M$ to be oriented.
\end{rem}


\bibliographystyle{apalike}
\bibliography{}

\end{document}