\documentclass[12pt,a4paper,leqno,twoside,report]{amsart}

\textwidth=154mm
\hoffset=-14mm
\usepackage{latexsym}
\usepackage{verbatim}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
%\usepackage[czech]{babel}
\usepackage[utf8]{inputenc}


\theoremstyle{plain}
\newtheorem{thm}{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ap}[thm]{Appendix}
\newtheorem{con}[thm]{Conjecture}

\theoremstyle{definition}
\newtheorem{Def}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{prob}[thm]{Problem}
\newtheorem{ex}[thm]{Exercise}

\def\N{\mathbb N}
\def\Z{\mathbb{Z}}
\def\Z2{\mathbb Z_2}
\def\R{\mathbb R}
\def\RP{\mathbb RP}

\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\S{\mathcal{S}}
\def\T{\mathcal{T}}
\def\L{\Lambda}
\def\d{\delta}

\def\df{\partial_\tau ^{F \ot B}}
\def\dg{\partial_\tau ^{G \ot B}}
\def\bo{\partial} %hranice

\def\bD{\bar{\Delta}}
\def\bW{\bar{W}}
\def\t{\tau}
\def\ot{\otimes}
\def\oth{\mathbin{{\otimes}_\t}}
\def\dif{\partial ^{F \ot B}}
\def\dig{\partial ^{G \ot B}}

\def\r{\rho}


\def\cb{C_* (B)}
\def\ce{C_* (E)}
\def\cg{C_* (G)}
\def\cf{C_* (F)}


\def\ep{\varepsilon}
\def\di{\partial}
\def\co{\circ}
\def\lra{\Leftrightarrow}
\def\ra{\Rightarrow}
\def\la{\Leftarrow}
\input xy
\xyoption{all}

\begin{document}


\title{Homework 8 -- 2017}
%\date{\today}
\maketitle

\bigskip

\begin{ex}
Let $f:M\to N$ be a map between two oriented compact manifolds of dimension $n$ with fundamental classes $[M]$ and $[N]$, respectively. We say that $f$ has degree $d$ if 
$$f_*([M])=d[N].$$
Prove that for every oriented compact manifold $M$ of dimension $n$ there is a map
$f:M\to S^n$ of degree 1. 
\end{ex}

\begin{ex}  Use cup product and $\Z2$ coefficients to show that $\RP^3$ is not homotopy equivalent to $\RP^2\vee S^3$.
\end{ex}

\end{document}