\documentclass[12pt]{amsart}
\setlength{\textheight}{9.75truein}
\setlength{\textwidth}{6.5truein}
\setlength{\evensidemargin}{0truein}
\setlength{\oddsidemargin}{0truein}
\setlength{\topmargin}{-.75truein}
%\parskip\baselineskip
\parskip10pt
\parindent0in
\usepackage{amssymb,latexsym,amscd}
\usepackage{graphicx}
\begin{document}
\begin{center}{\bf Homework 8, due Tuesday, November 27, 2012}\end{center}
%\vskip\baselineskip
Do any 4 of the 6 problems.
Each problem is worth 25 points. Solutions will be graded for correctness, clarity and style.
\vskip-.5in
\begin{itemize}
\item[(1)] Let $X$ be a topological space and let $Y$ be a set. Let $f:X\to Y$ be a map,
not necessarily surjective. Let ${\mathcal T}_Y=\{V\subseteq Y:f^{-1}(V) \text{ is open in }X\}$.
Show that ${\mathcal T}_Y$ defines a topology on $Y$. (In class we used this construction
mainly when $f$ is surjective.)
\vskip3pt
\item[(2)] Let $X$ be a topological space and let $Y$ be a set. Let $f:X\to Y$ be a map
such that $f(X)$ is a single point $p\in Y$.
Let ${\mathcal T}_Y=\{V\subseteq Y:f^{-1}(V) \text{ is open in }X\}$.
Show that ${\mathcal T}_Y$ is the discrete topology on $Y$.
\vskip3pt
\item[(3)] Let $\sigma^k$ be the standard $k$-simplex, so
$\sigma^k=\{(x_1,\ldots,x_{k+1})\in{\bf R}^{k+1}: x_1\geq 0,\ldots,x_{k+1}\geq0, x_1+\cdots+x_{k+1}=1\}$.
Let $\tau^k$ be a $k$-simplex in ${\bf R}^N$, so $\tau^k=\langle v_1,\ldots,v_{k+1}\rangle$ where the points
$v_i\in {\bf R}^N$ are geometrically independent. Define a map $f:\sigma^k\to\tau^k$ by $f((x_1,\ldots,x_{k+1}))=\sum_ix_iv_i$.
(a) Show that $f$ is bijective.
(b) Assuming that $f$ is continuous, show that $f$ is a homeomorphism. This shows that
all $k$-simplices are homeomorphic to the standard $k$-simplex and hence to each other.
[Aside: $h:{\bf R}^{k+1}\to{\bf R}^N$ defined by $f((a_1,\ldots,a_{k+1}))=\sum_i a_iv_i$ is a
linear transformation. It is known and not hard to show that linear transformations
are continuous under the standard topologies, basically because a linear transformation just involves addition
and multiplication of the variables by constants, and these are continuous. Since $f$ is the restriction of $h$ to $\sigma^k$,
this means that $f$ is indeed continuous.]
\vskip3pt
\item[(4)] Let $0\leq r\leq k\leq K$ where $r$, $k$ and $K$ are integers.
Give a formula for the number of $k$-simplices contained in a
$K$-simplex which contain a given $r$-simplex.
[For example, if $r=0$, $k=1$ and $K=3$, this is asking how many edges
of a tetrahedron contain a given vertex of the tetrahedron, so your formula should give 3.]
\vskip3pt
\item[(5)] Consider a 14-gon. Go around the circumference of the 14-gon, labeling the edges
in turn as follows: $a$, $b$, $a^{-1}$, $c$, $b^{-1}$, $c^{-1}$, $d$, $e$, $g$, $e^{-1}$, $f$, $g^{-1}$, $f^{-1}$, $d^{-1}$.
This gives a planar diagram, with labels and exponents, for a multi-holed torus as in the diagram below at left. (The planar diagram can also be shown with labels and arrows as in the diagram below at right. Think of $a$ as specifying an arrow on the
corresponding edge of the 14-gon pointing in the direction in which you're going around the 14-gon.
Think of $a^{-1}$ as specifying an arrow on the
corresponding edge of the 14-gon but pointing in the direction opposite to
which you're going around the 14-gon.) Determine the number of holes of the torus you get
by making the identifications specified by the labels.
\vskip-6pt
\vbox to0in{\hskip1in\includegraphics[width=1.5in]{14gonA.pdf}}
\vskip0pt\hskip3in\includegraphics[width=1.5in]{14gon.pdf}
\vskip6pt
\item[(6)] Determine whether the points $(1,1,1),(2,3,4),(3,5,7)\in{\bf R}^3$ are geometrically independent.
\end{itemize}
\end{document}