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\newcommand{\myTitle}{HW 9, due Friday, April 12}
\newcommand{\myClass}{Math 411}
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{\large \myClass}\\[5pt]
\myTitle
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\begin{enumerate}
\item Let $V$ be an oriented vector space with scalar product
  $\langle\ ,\ \rangle$ of index $s$.  Let $e_1,\dots,e_n$ be an orthonormal
  basis for $V$ with $\langle e_i,e_i\rangle=\varepsilon_i\in\{-1,1\}$ for each
  $i$, and let $\omega_V$ be the volume form for $V$.  For each $k$, let
  $*\colon\Lambda^kV^*\to\Lambda^{n-k}V^*$ be the $*$-operator.
\begin{enumerate}
 \item Compute $*1$ and $*\omega_V$.
   \medskip

   {\sc solution:}

 \item Show $**=(-1)^{k(n-k)+s}\,\mbox{Id}_{\Lambda^kV^*}$. (Recall that if
   $e_1,\dots,e_n$ is a positively oriented orthonormal basis for $V$, then
   we have an orthonormal basis $\{e^{*}_{\mu}\}$ for $\Lambda^kV^*$ with
   $\langle
   e^{*}_{\mu},e^{*}_{\gamma}\rangle=\delta_{\mu\gamma}\varepsilon_{\mu}$
   where $\varepsilon_{\mu}=\prod_{i}\varepsilon_{\mu_i}$. We also have
   $\mathrm{sgn}(\tau_{\mu})$ defined by
   \[
   e^{*}_{\mu}\wedge e^{*}_{\tilde{\mu}}=\mathrm{sgn}(\tau_{\mu})\,\omega_V
   =\mathrm{sgn}(\tau_{\mu})\,e^{*}_1\wedge\cdots\wedge e^{*}_{n},
   \]
   where
   $\tilde{\mu}$ is the complement of $\mu$ in $\{1,\dots,n\}$. Then
   $*e^{*}_{\mu}=\varepsilon_{\mu}\,\mathrm{sgn}(\tau_{\mu})\,e^{*}_{\tilde{\mu}}$.)
   \medskip

   {\sc solution:}

 \item Show $\langle *\eta,*\zeta\rangle=(-1)^s\langle\eta,\zeta\rangle$ for all
   $\eta,\zeta\in\Lambda^kV^*$. (Try to do this without appealing to the
   description of the $*$-operator in terms of a basis.  Use defining properties
   of the $*$-operator, instead.)
   \medskip

   {\sc solution:}

 \item For $\eta,\zeta\in\Lambda^kV^*$, find the relationship between
    (i) $*(\eta\wedge *\zeta)$, (ii) 
$*(*\eta\wedge\zeta)$, (iii) $*(\zeta\wedge *\eta)$, and (iv) $*(*\zeta\wedge\eta)$ by relating them all to $\langle\eta,\zeta\rangle$.
   \end{enumerate}
   \medskip

   {\sc solution:}

  \item Let $V=\R^4$ with standard oriented basis $e_1,\dots,e_4$ and with the
    Lorentzian scalar product: 
\[
\langle e_i,e_j\rangle =
\left\{
\begin{array}{rl}
0&\quad\mbox{if $i\neq j$},\\
1&\quad\mbox{if $i=j$ and $i\neq 4$},\\
-1&\quad\mbox{if $i=j=4$}.
\end{array}
\right. 
\]
For each $k=0,1,2,3,4$, describe the $*$-operator by showing what it
  does to the standard basis for $\Lambda^kV^*$.  (For instance,
\[
e_1^*\wedge e_2^*\wedge e_3^*,\quad e_1^*\wedge e_2^*\wedge
  e_4^*,\quad e_1^*\wedge e_3^*\wedge e_4^*,\quad e_2^*\wedge
  e_3^*\wedge e_4^*
\]
is the standard basis for $\Lambda^3V^*$. Order the indices lexicographically,
as above.) 
   \medskip

   {\sc solution:}

\item Now let $V=\R^3$ with the standard inner product and orientation.  What is the
  precise relation between the $*$-operator and the usual cross product (with
  proof)?
   \medskip

   {\sc solution:}

\item What are the cohomology groups of a solid donut?
   \medskip

   {\sc solution:}

\item What are the cohomology groups of (the surface of) a donut with $g$ holes?
  (Recall that in the last homework, you computed the cohomology of a punctured
  single-holed donut.)  You may use the fact that if $M$ is a compact,
  connected, orientable $n$-manifold with $n\geq2$,  and $p\in M$,
  then $H^n(M\setminus\{p\})=0$.
   \medskip

   {\sc solution:}

\item Consider the cone, $\sigma$, in $\R^2$ generated by the vectors $(0,1)$ and
  $(3,-1)$.  Describe the (affine) toric variety, $U_{\sigma}$.
   \medskip

   {\sc solution:}

\item Each of the quadrants of $\R^2$ forms a cone.  Consider the fan,
  $\Delta$, consisting of
  these four cones.  Describe the toric variety $X(\Delta)$ by identifying the
  affine toric varieties corresponding to each cone and describing the gluing
  instructions.  Extra: Can you identify $X(\Delta)$ as a well-known manifold?
   \medskip

   {\sc solution:}

\end{enumerate}
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