2019-01-30-intro-homology.tex

\documentclass[11pt]{amsart}
\usepackage[utf8]{inputenc}
\usepackage{fullpage}
\usepackage{ccg-macros}

% pandoc transfers h2 to subsections, I prefer h2 for sections
\let\subsubsection\subsection
\let\subsection\section
\let\section\chapter
\let\chapter\part
% section numbering
\setcounter{secnumdepth}{5}
% urls as footnotes
\usepackage[unicode=true]{hyperref}
\renewcommand{\href}[2]{#2\footnote{\url{#1}}}
% fix: parskip mangles ams table of contents
\usepackage{parskip}
\makeatletter
\renewcommand\tableofcontents{%
    \@starttoc{toc}%
}
\makeatother
\providecommand{\tightlist}{%
  \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}

\title{Introduction to Homology}
\author{Colton Grainger (MATH 6220 Topology 2)}
\date{2019-01-14}
\thanks{\emph{Git Repo:} \url{https://github.com/coltongrainger/fy19top2}}

\begin{document}

\maketitle

\subsection{Problems due 2019-01-30}

\subsubsection{}

Let \(X = \bigcup_{i\in I} X_i\) where the \(X_i\) are the (disjoint)
path components of \(X\). Prove that
\[H_*(X) \cong \bigoplus_{i\in I} H_*(X_i).\]

\subsubsection{}

Let \(X\) and \(Y\) be path connected spaces.

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Prove that any map \(f \colon X \to Y\) induces an isomorphism
  \(f_* \colon H_0(X) \xrightarrow{\cong} H_0(Y)\).
\item
  Prove that any map \(f \colon X \to X\) induces the identity on
  \(H_0(X)\).
\end{enumerate}

\subsubsection{}

Use the Hurewicz theorem to solve the following problems:

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Compute \(H_1(K)\) for the Klein bottle \(K\).
\item
  Compute \(H_1\) of \(X=\prod_{j \in J} X_j\) for topological spaces
  \(X_j\) in terms of \(H_1(X_j)\).
\item
  Let \(X_i\) with base points \(x_i \in X_i\). Suppose that there are
  open sets \(x_i\in U_i \subseteq X_i\) such that \(x_i\) is a
  deformation retract of \(U_i\). Show that
  \[{H}_1 \left( \bigvee_{i\in I}  X_i \right)  \cong \bigoplus_{i\in I} {H}_1(X_i).\]
\end{enumerate}

\subsubsection{}

Let \(f \colon X \to Y\) be a covering space of path connected spaces
with \(f(x_0) = y_0\). By the fundamental theorem of covering spaces,
\(f_{\#} \colon \pi_1(X,x_0) \to \pi_1(Y, y_0)\) is a monomorphism. Is
\(f_* \colon H_*(X) \to H_*(Y)\) also a monomorphism. Prove that it is,
or give a counterexample.

\end{document}