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\lhead{\tiny Math in Moscow, HSE, Spring 2013} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny Test assignment \#2,\ \ \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{1}{Test assignment \#2}

{\scriptsize
{\bf Rules:} Please solve this in class, before 21:30
February 04, 2013, and give me the written solutions.
The score for the test is computed using the formula
$s= 3p- (\max(p-4,0))$. Results will be
announced at \url{http://bogomolov-lab.ru/KURSY/GEOM-2013/}
}

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\exercise
Consider {\bf the Moebius strip} $M$ as a quotient space of
$\R \times [0,1]$ with opposite lines glued together 
with reverse orientation. Construct a closed embedding
of $M$ to $\R^4$.
\ez

\exercise[4 points] Construct a closed embedding
of Moebius strip to $\R^3$, or prove that it does not exist.
\ez

\exercise
Construct an embedding of $(S^1)^3$ to $\R^4$, or prove that
it does not exist.
\ez

\exercise
Construct an embedding of $S^1\times S^2$ to $\R^4$, or prove that
it does not exist.
\ez

\exercise
Let $M$ be an $n$-dimensional manifold.
Construct a smooth, surjective map from $M$ to the torus
$(S^1)^n$.
\ez



\exercise
Let $R$ be a ring of continuous $\R$-valued functions
on a topological space $M$, and $I\subset R$ a prime ideal.
Prove that $I^2=I$.
\ez

\exercise[3 points]
Let $R$ be a ring of continuous $\R$-valued functions
on a compact topological space $M$, and $I\subset R$ an ideal.
Prove that there exists $Z\subset M$ such that $I$ is an 
ideal of all functions vanishing at $Z$, or find a 
counterexample.
\ez


\exercise[3 points]
Let $R$ be a ring of germs of continuous $\R$-valued functions
at a point $x\in M$ of a manifold $M$, and $I\subset R$ a prime ideal.
Prove that $I$ is maximal, or find a counterexample.
\ez

\definition
A sheaf ${\cal B}$ is called {\bf flasque} if any restriction map
${\cal B}(U)\arrow {\cal B}(V)$ is surjective.
\ed


\exercise
 Prove that any 
flasque sheaf is soft.
\ez


\exercise
For a given sheaf ${\cal B}$, find a sheaf monomorphism
${\cal B}\hookrightarrow {\cal B}'$ to a flasque sheaf.
\ez

\definition
A sheaf ${\cal I}$ is called {\bf injective} if
for any sheaf morphism  ${\cal B}\stackrel \phi\arrow {\cal I}$
and a monomorphism ${\cal B}\hookrightarrow {\cal B}'$,
the map $\phi$ can be extended to a morphism
${\cal B}'\stackrel \phi\arrow {\cal I}$.
\ed

\exercise Prove that any injective sheaf is flasque.
\ez 



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