Preliminary material

In fact this first version of these notes is just a hastily constructed pastiche of this material which we will improve on (and expand) during the semester.

It is assumed that the student has had an exposure to topology, including at least basic definitions and a few examples in algebraic topology such as the fundamental group and homology groups.

In the next Chapter we introduce the idea of manifold as a subset of ${ \bf R^n}$ .For completness, here are some basic definitions for refenence as well as to establish notations:

We define n-dimensional space, denoted ${ \bf R^n}$ , to be the set of all ordered n-tuples of real numbers, together with a notion of distance, $\delta$ . If $p=(x_1, x_2,\ldots,x_n)$ and $q=(y_1, y_2,\ldots, y_n)$ are two points of ${ \bf R^n}$ , the distance between these points denoted by $\delta (p,q)$ and defined by $\delta (p,q) = \sqrt{\sum_{i=1}^n (x_i-y_i)^2}$

Let $p \in {\bf R^n}$ , and $\epsilon$ be a given number, $\epsilon \gt 0$ . Then the open n-dimensional ball in ${ \bf R^n}$ with center p and radius $\epsilon$ is defined to be the set $\{ q \in {\bf R^n} \mid \delta (p$ , $q) < \epsilon \}$ . We denote this set by $N_\epsilon (p)$ . We also refer to $N_\epsilon (p)$ as an $\epsilon$ -neighborhood of p in ${ \bf R^n}$ .

Let X be a subset of ${ \bf R^n}$ with $p \in X$ . Given a number $\epsilon$ , with $\epsilon \gt 0$ , we define an $\epsilon$ -neighborhood of p in X to be the subset of X, $N_\epsilon (p$ , $X) = N_\epsilon (p) \cap X$ .

If $p \in {\bf R^n}$ and $\epsilon$ is a number, $\epsilon \gt 0$ , then the closed n-dimensional ball with center p and radius $\epsilon$ is defined to be $\{ q \in {\bf R^n} \mid \delta (p, q) \leq \epsilon \}$ . We denote this set by $\overline{N}_{\epsilon}(p)$ , or sometimes, $B_\epsilon (p)$ .

Let $a \mbox{ and } b$ be numbers with a < b. The following subsets are all called intervals:

Let $A \subseteq X \subseteq {\bf R^n}$ . We say that A is dense in X if every open subset of X contains a point of A.

Let Q denote the rational numbers, and Z denote the irrational numbers. Then Q and Z are dense in ${ \bf R^1}$ . $\diamondsuit$

It is sometimes useful to note that any subset of ${ \bf R^n}$ has a countable dense subset. Let in ${ \bf R^n}$ , let $\mathcal G$ be the set of all open neighborhoods, $N_\epsilon (p)$ where $\epsilon$ is a rational number and each of the n coordinates of p is a rational number; $\mathcal G$ will have countably many elements. For each $N_\epsilon (p) \in \mathcal G$ , chose one point, call it $x^p_\epsilon$ , of $X \cap N_\epsilon (p)$ , (if this set is non-empty). Then $\{ x^p_\epsilon \}$ will be a countable subset of X, and one can verify that it is a dense subset of X. $\diamondsuit$

The definition of limit point is not completely standard. Here is what we will use:

Let $A \subseteq X \subseteq {\bf R^n}$ , $x \in X$ . We is say that x is a limit point of A in X if every open subset, U, of X which contains x contains a point of $A - \{x\}$ . (That is, U contains a point of A other than x.)

**Figure 0-1:** On the left the Euclidian half-plane, R²₊, see Definition 0.10. On the right the open Euclidian half-plane, $\stackrel {\circ}{R^2_+}$ , see Definition 0.11. Note that R²₊ contains points of the y-axis but $\stackrel {\circ}{R^2_+}$ does not.
$\begin{figure} \begin{center} \bigskip {\epsfxsize=2.0in \leavevmode\epsffile{figBChap2.eps}}\end{center}\end{figure}$

We define n-dimensional half space to be:
$R^n_+ = \{(x_1,x_2, \ldots,x_n)\in {\bf R^n} \mid 0 \leq x_1 \})$ . (See Figure 0-1).

We define open n-dimensional half space to be:
$\stackrel {\circ}{R^n_+} = \{(x_1, x_2, \ldots,x_n)\in {\bf R^n} \mid 0 < x_1 \})$ . (See Figure 0-1).

For certain subsets of ${ \bf R^n}$ , generalized cylindrical coordinates may be more convenient. Here we locate points of ${ \bf R^n}$ using polar coordinates in the first two coordinates, Cartesian coordinates for the rest.

Cylindrical coordinates for $(x_1, x_2, \ldots, x_n) \in { \bf R^n}$ are $(r, \theta, x_3, \ldots, x_n)$ where $r = \sqrt{x_1^2 + x_2^2} \mbox{ and } \theta = \arctan (x_2/x_1) \mbox{ if }x_1 \neq 0$ .

As with polar coordinates, points have multiple descriptions such as $(1,0,2,3) = (1,2 \pi,2,3)$ and (0,0,2,3) = (0,1,2,3).

**Figure 0-2:** Eight pages in ${ \bf R^3}$ , see Example 0.14
$\begin{figure} \begin{center} \bigskip {\epsfxsize=2.0in \leavevmode\epsffile{pages.eps}}\end{center}\end{figure}$

For an angle, $\alpha$ , the subset $H^{n-1}_\alpha$ is defined, using generalized cylindrical coordinates, $H^{n-1}_\alpha = \{ (r, \theta, x_3, \ldots, x_n) \in { \bf R^n} \mid \theta = \alpha\}$ . Such a subset is called an (n-1)-dimensional page of ${ \bf R^n}$ .

Figure 0-2 we indicate eight pages , H²₀, $H^2_{\pi/4}$ , $H^2_{\pi/2}$ , $H^2_{3 \pi/4}$ , $H^2_{\pi}$ , $H^2_{5 \pi/4}$ , $H^2_{3\pi/2}$ , and $H^2_{7 \pi/4}$ in ${ \bf R^3}$ . Only a square on each page is shown. The term ``page'' comes from the following image--if we take a book and open all the way until the front cover meets the back cover the pages of the book will fan out and look something like Figure 0-2. $\diamondsuit$

We define the n-dimensional sphere to be the subset $S^n = \{\vec{x} \in {\bf R}^{n+1} \mid \delta (\vec{x}, \vec{0}) = 1 \}$ where $\vec{0}$ denotes the origin of ${\bf R}^{n+1}$ . (See Figure 0-3).

**Figure 0-3:** On the left is $S^0 \subseteq \bf R^1$ . It consists of the two numbers 0 and 1. In the middle is $S^1 \subseteq \bf R^2$ . It is the unit circle in the plane. On the right is $S^3 \subseteq \bf R^3$ . It is a surface, the unit sphere. See Definition 0.15.
$\begin{figure} \begin{center} \bigskip {\epsfxsize=3.0in \leavevmode\epsffile{threeSpheres.eps}}\end{center}\end{figure}$

The standard n-ball ( sometimes called the unit n-ball) in ${ \bf R^n}$ is the set $D^n = \{ \vec{x} \in { \bf R^n} \mid \delta (\vec{x}, \vec{0}) \leq 1 \}$ . (See Figure 0-4).

**Figure 0-4:** On the left is $D^0 \subseteq \bf R^0$ . It is a single point. In the middle-left is $D^1 \subseteq \bf R^1$ . It is the interval [-1, 1] in the line. On the middle-right is $D^2 \subseteq \bf R^2$ . It is a disk whose edge is the unit circle. On the right is $D^3 \subseteq \bf R^3$ . It is solid ball consisting of all points on the unit sphere and also those inside it, see Definition 0.16. Unfortunately there is no simple graphical method to distinguish a drawing of S², as in Figure 0-3, from a drawing of D³.
$\begin{figure} \begin{center} \bigskip {\epsfxsize=3.0in \leavevmode\epsffile{fourBalls.eps}}\end{center}\end{figure}$

So, S¹ is the unit circle in the plane; S² is the standard unit sphere in space. Also, S⁰ consists of two points-the numbers -1 and 1 of ${ \bf R^1}$ , (see Figure 0-3).

Also, D¹ is the closed interval [-1, 1] in ${ \bf R^1}$ , D² is the unit disk in the plane, D³ the unit ball in ${ \bf R^3}$ , see Figure 0-4 and, for any n, $S^n \subseteq D^{n+1}$ . $\diamondsuit$

The standard n-simplex in ${\bf R}^{n+1}$ , denoted $\Delta ^n$ , is defined by $\Delta ^n = \{(x_1, x_2, \ldots,x_{n+1})\in {\bf R}^{n+1} \mid$ $x_1+ \cdots + x_{n+1} = 1$ with $0 \leq x_i$ for all i, $1 \leq i \leq n+1$ $\}$ .

**Figure 0-5:** On the left, the standard 1-simplex, $\Delta^1$ , a subset of ${ \bf R^2}$ . On the right, the standard 2-simplex, $\Delta^2$ , a subset of ${ \bf R^3}$ . See Definition 0.18
$\begin{figure} % latex2html id marker 526 \begin{center} \bigskip {\epsfxsize=2.0in \leavevmode\epsffile{simplexes.eps}}\end{center}\end{figure}$

standard 1-simplex in the plane is a line segment with endpoints (1,0) and (0,1). It is the portion of the line with equation x₁ + x₂ = 1 which lies in the first quadrant of the plane where $0 \leq x_1$ and $0 \leq x_2$ . The standard 2-simplex in space is a (two-dimensional) triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1), since it is the portion of the plane x₁ + x₂ + x₃ = 1 contained in the first octant of three-dimensional space, (see Figure 0-5).

**Figure 0-6:** Figure for stereographic projection of a 2-sphere mapping minus the ``south pole'', homeomorphically, to the x-y-coordinate plane, see Proposition 0.20.
$\begin{figure} \begin{center} \bigskip {\epsfxsize=2.0in \leavevmode\epsffile{stereo.eps}}\end{center}\end{figure}$

Let Sⁿ be the standard sphere in ${\bf R}^{n+1}$ and let $P \in S^n$ be the point $P = (0, 0, \ldots , 0, -1)$ . Then $S^n -\{P\}$ is homeomorphic to ${ \bf R^n}$ .

In fact this homeomorphism can be described as follows: let $x \in S^n -\{P\}$ and let l_x be the line in ${\bf R}^{n+1}$ containing x and P. Identify ${ \bf R^n}$ as a coordinate hyperplane of ${\bf R}^{n+1}$ as its image, under the standard inclusion map. Then h(x) is defined to be the intersection of the line l_x and the hyperplane ${ \bf R^n}$ . The map h is called stereographic projection.

Proof: Suppose $x \in S^n$ , $x = (x_1, x_2, \ldots , x_n, x_{n+1} )$ .In vector notation we can write l_x as $l_x = \vec{P} + t (\vec{x} -\vec{P})$ . In term of coordinates this is:

$\begin{displaymath} (t x_1, t x_2, \ldots, t x_n, -1 + t ( x_{n+1} +1) ).\end{displaymath}$

The intersection we want is the point on l_x, where the last coordinate is zero. This happens if $t = \frac{1}{ x_{n+1} +1}$ .

$\begin{displaymath} h((x_1, x_2, \ldots , x_n, x_{n+1} )) = ( \frac{x_1}{ x_{n+1... ...\frac{x_2}{ x_{n+1} +1}, \ldots , \frac{x_n}{ x_{n+1} +1}, 0 ) \end{displaymath}$

In the next definitions, we use vector notation for ${ \bf R^n}$ . If M is an $m \times n$ matrix $M \vec{x}$ is matrix multiplication where we write the m-dimensional vector, $\vec{x}$ , as a single column matrix. Also |M| denotes the determinant of a square matrix..

If M is an $m \times n$ matrix and $\vec{B}$ is an n-dimensional vector, we can define a function $F:R^m \to R^n$ by $F(\vec{x}) = M \vec{x} + \vec{B}$ . Such a function is called an affine transformation or affine map. In the case that m=n, and $\vert M\vert \neq 0$ , then we say that F is a non-singular affine map.

An affine map with $\vec{B} = \vec{0}$ is called a linear transformation or linear map. A non-singular affine map with $\vec{B} = \vec{0}$ is called a non-singular linear transformation or non-singular linear map

An affine map $f \colon { \bf R^1} \to { \bf R^1}$ is simply a function of the form f(x) = m x +b; f is non-singular means $m \neq 0$ .

A critical definition we use here, common in linear algebra, is that of an orthogonal matrix. Here M^T denotes the transpose of M, that is if M = [a_{i j}], then M^T = [a_{j i}].

Let M be an $n \times n$ matrix. We say M is an orthogonal matrix if M^T = M^-1.

If M is an orthogonal matrix, then M^-1 is also orthogonal. If M is orthogonal, then M^T = M^-1. Take the transpose of both sides of this equation, and we get M = (M^-1)^T. Since M is the inverse of M^-1, this last equation can be written (M^-1)^-1 = (M^-1)^T, which verifies that M^-1 is orthogonal.

If M is an orthogonal matrix, then the determinant of M is +1 or -1. We have M^T M = I. But $\det{M^T} = \det{M}$ , so taking the determinant of both sides of the previous equation, and since the determinant of the product is the product of the determinants, we get $\det{M} \det{M} = 1$ , thus $\det{M} = \pm 1$ . $\diamondsuit$

An affine map $F:R^n \to R^n$ defined by $F(\vec{x}) = M \vec{x} + \vec{B}$ is called a rigid motion of ${ \bf R^n}$ , if M is an orthogonal matrix.

If $M = \left( ^{ \ \ \cos (\theta)\ \sin (\theta)}_{- \sin (\theta)\ \cos (\theta)} \right)$ , $x \in R^2$ , function $F(\vec{x}) = M \vec{x}$ corresponds to a counter-clockwise rotation of angle $\theta$ about the origin. The inverse of M corresponds to clockwise rotation of angle $\theta$ , with matrix $M^{-1} = \left( ^{ \ \ \cos (-\theta)\ \sin (-\theta)}_{- \sin (-\theta)\ \cos ... ...cos (\theta)\ -\sin (\theta)}_{ \sin (\theta)\ \ \ \cos (\theta)} \right) = M^T$ . Thus M is orthogonal.

The matrix $\left( ^{ 1\ ~~0}_{ 0\ -1} \right)$ corresponds to reflection in the x-axis, while $\left( ^{ -1\ 0}_{~~0\ 1} \right)$ corresponds to reflection in the y-axis.

$\begin{displaymath} R_z(\theta) = \left( \begin{array} {ccc} \cos (\theta) & \s... ...\theta) & \cos (\theta) & 0 \ 0 & 0 & 1 \end{array} \right) \end{displaymath}$

In the term ``rigid motion'' ``rigid '' refers to the fact is that distances are preserved as the next proposition shows.

Suppose we have a rigid motion, F, in ${ \bf R^n}$ . If $\vec{x} \mbox{ and } \vec{y}$ are two vectors in ${ \bf R^n}$ then $\delta(\vec{x}, \vec{y} ) = \delta(F(\vec{x}), F(\vec{y})).$

$\begin{displaymath} \delta(\vec{x}, \vec{y} ) = \sqrt{(\vec{x} -\vec{y}) \cdot (... ...} -\vec{y})} = \sqrt{(\vec{x} -\vec{y})^T (\vec{x} -\vec{y})} ,\end{displaymath}$

$\begin{displaymath} (\delta(F(\vec{x}), F(\vec{y})))^2 = \left( F(\vec{x})- F(\vec{y})\right) \cdot \left( F(\vec{x}) - F(\vec{y}) \right) = \end{displaymath}$

$\begin{displaymath} ((M \vec{x} + \vec{B}) - (M \vec{y} + \vec{B})) \cdot ((M \vec{x} + \vec{B}) - (M \vec{y} + \vec{B})) = \end{displaymath}$

$\begin{displaymath} M (\vec{x} - \vec{y} ) \cdot M (\vec{x} - \vec{y} ) = (M (\vec{x} - \vec{y} ))^T M (\vec{x} - \vec{y} ) = \end{displaymath}$

$\begin{displaymath} ( (\vec{x} - \vec{y} )^T M^T ) M (\vec{x} - \vec{y} ) = (\vec{x} - \vec{y} )^T (M^T M) (\vec{x} - \vec{y} ) = \end{displaymath}$

$\begin{displaymath} (\vec{x} - \vec{y} )^T I (\vec{x} - \vec{y} ) = (\vec{x} - \... ...} ) \cdot (\vec{x} - \vec{y} ) = (\delta(\vec{x}, \vec{y}))^2.\end{displaymath}$

Taking square roots it follows that $\delta(F(\vec{x}), F(\vec{y}) = \delta(\vec{x}, \vec{y})$ . width4pt height6pt depth1.5pt

Suppose $X \subseteq { \bf R^n} \subseteq { \bf R^{n+1}}$ and let $p \in {\bf R^n}$ be a point whose last coordinate is non-zero. Then the cone on X from p is the union of all line segments in ${\bf R}^{n+1}$ from p to a point of X.

Suppose $X \subseteq { \bf R^n} \subseteq { \bf R^{n+1}}$ and let $p, q \in { \bf R^n}$ be a points with the last coordinate of p is positive the last coordinate of q is negative. Then the suspension of X from p and q is the union of all line segments in ${\bf R}^{n+1}$ from p to a point of X and from q to a point of X.

we consider $S^1 \subseteq { \bf R^2} \subseteq { \bf R^3}$ , then the cone from any point is, geometrically, a cone. $\diamondsuit$

If we have $A \subseteq X \subseteq {\bf R^n}$ , the concept of closure gives us a way of associating to A a (possibly larger) closed subset of X. The next definition is a way of associating to A a (possibly smaller) open subset of X.

Suppose $a \in A \subseteq X \subseteq { \bf R^n}$ ; we say a is an interior point of A in X, if there is an open subset, U, of X with $a \in U \subseteq A$ .

The subset of A consisting of all interior points of A is called the interior of A. This is denoted by int(A,X). As with closure, if the set X is understood in context, one often uses the notation int(A).

Open subsets can be defined in terms of interior points. It follows from Proposition 0.32 that if $U \subseteq X \subseteq { \bf R^n}$ , U is an open subset of X if and only if every point of U is an interior point of U. $\diamondsuit$

Suppose $a \in A \subseteq X \subseteq { \bf R^n}$ ; we say a is a boundary point of A in X if $a \in \overline{A} \cap \overline{X - A}$ . (Note: the closures are closures in X.)

The set of boundary points of A in X is called the boundary of A in X, denoted Bd(A).

If $A \subseteq X \subseteq {\bf R^n}$ , then the boundary of A is a closed subset of X, since it is the intersection of closed subsets. $\diamondsuit$

If $A \subseteq X \subseteq {\bf R^n}$ , then the interior of A and the boundary of A are disjoint sets, whose union is A. width4pt height6pt depth1.5pt

Here are a few basic facts relating closure, boundary and interior of a subset to Cartesian products.

Suppose $A \subseteq X \subseteq {\bf R^n}$ and $B \subseteq Y \subseteq { \bf R^m}$ then

If A and B are closed subsets then $A = Cl_X (A) \mbox{ and } B = Cl_Y (B)$ and so the first formula of Proposition 0.37 c) simplifies to the more easily remembered:

$\begin{displaymath} Bd(A \times B) = \big( Bd(A) \times B \big) \cup \big( A \times Bd(B)\big) .\end{displaymath}$

**Figure 0-7:** An example of the boundary of the product $D^2 \times I$ , see Example 0.39. On the upper left is shown all of $Bd(D^2 \times I)$ , the upper right shows $Bd(D^2) \times I$ , and at the lower left is shown $D^2 \times Bd(I)$
$\begin{figure} \begin{center} \bigskip {\epsfxsize=2.0in \leavevmode\epsffile{cylinder.eps}}\end{center}\end{figure}$

The formulas of Proposition 0.37 c) would benefit from an example. Suppose A is the closed unit disk in ${ \bf R^2}$ and B is the unit interval in ${ \bf R^1}$ . Then Bd(A) is the unit circle and Bd(B) is a two point set consisting of the two numbers 0 and 1.

Since A and B are closed subsets, we may use the simplified formula of Remark 0.38.

$A \times B$ is a cylindrical solid in ${ \bf R^3}$ and $Bd(A \times B)$ is the ``surface of the cylindrical solid'', see Figure 0-7, upper left. Note $Bd(A \times B)$ is the union of the cylindrical surface, $Bd(A) \times B$ , which is the ``side of the solid cylinder'', Figure 0-7, upper right, and $A \times Bd(B)$ , the two disks comprising the top and bottom , Figure 0-7, lower left. The intersection $Bd(A) \times B \cap A \times Bd(B)$ consists of two disjoint circles. $\diamondsuit$

Suppose $X \subseteq {\bf R^n}$ , $f\colon I \to X$ and $g\colon I \to X$ are paths in X with f(1) = g(0). Define a path $f\ \ast \ g$ in X called the concatenation of f and g by

$\begin{displaymath} % latex2html id marker 1332 f \ast g (t) = \left\{ \begin{ar... ... & \mbox{if $\frac{1}{2} \leq t \leq 1$} \end{array} \right. .\end{displaymath}$

This final example of this Chapter, especially for the irrational angle case, is an interesting, non-trivial one. In a future Chapter, we will re-state the problem with a new perspective, Example 0.44.

Let C denote the unit circle in the plane, and let $X_\theta$ be the set of points of C corresponding to all multiples of a given angle $\theta$ . That is, write $x_\theta = (\sin(\theta), \cos(\theta))$ and let $X_\theta = \{x_{n \theta } \}_{n \in N}$ , N the natural numbers.

We first note that $X_\theta$ will be a finite set if and only if $\theta$ is a rational multiple of $2 \pi$ . Suppose $\theta$ is a rational multiple of $2 \pi$ and write $\theta = \frac{p}{q} 2 \pi$ , where p and q are positive integers, the fraction written in reduced form. Then $X_\theta$ will consist of q points: $X_\theta = \{\theta, 2 \theta, \ldots q \theta \}$ . Conversely, if $X_\theta$ is finite, there there must be a p and a q such that $p \theta = q \theta + N 2 \pi$ and $\theta = \frac{N}{p-q} 2 \pi$ .

Thus, if $\theta$ is an irrational multiple of $2 \pi$ , $X_\theta$ is infinite. We next claim that, in this case, $X_\theta$ is a dense subset of C. We first focus on the special case of x₀ = (1,0), and show that x₀ is a limit point of $X_\theta$ . We show that, for any $\epsilon \gt 0$ , there is a point, $x_\alpha$ , of $X_\theta$ whose distance from x₀ is less than $\epsilon$ .

Find an integer N so that $4/N < \epsilon$ . Divide C into N subarcs: $A_1, \ldots A_N$ where $A_i = \{ x_\theta \colon 2 \pi (i-1) \leq \theta \leq 2 \pi i \}$ . (The reason for using 4/N is to insure that distance between any two points of A_i is less than $\epsilon$ . A look at the case $\epsilon =1$ , N=4 will clarify this point. The number 4 is just a continent choice of a number greater than $\pi$ .) Since $X_\theta$ is infinite, we can find some k such that A_k contains two points, $p \theta$ and $q \theta$ , of $X_\theta$ .Consider the subset of $X_\theta$ , $Y_\theta = \{x_{n (p-q) \theta } \}_{n \in N}$ . Consecutive points of $Y_\theta$ are a distance less than $\epsilon$ apart, so one point, $x_\alpha$ , of $Y_\theta$ , must lie in A₁, thus $\delta (x_\alpha, x_0) < \epsilon$ .

We have shown that x₀ is a limit point of $X_\theta$ . There are a number of ways to show that any $x_\phi \in X_\theta$ is also a limit point. The simplest is to verify that, given any $\epsilon \gt 0$ , $\delta ( x_{\phi +\alpha}, x_\phi) < \epsilon$ . $\diamondsuit$

Suppose $X \subseteq {\bf R^n}$ , function $f \colon X \rightarrow X$ is continuous. We can derive an infinite set of continuous functions from X to itself, namely $\{f$ , $f\circ f$ , $f\circ f\circ f$ , ... $\}$ .

$f \colon S^1 \to S^1$ be rotation by an angle $\theta$ .We can restate the results of Example 0.41 as follows. Suppose $x \in S^1$ . If $\theta$ is a rational multiple of $2 \pi$ , then the orbit of x under f is a finite subset. If $\theta$ is a irrational multiple of $2 \pi$ , the orbit of x is dense in S¹. $\diamondsuit$