Polyhedral Subdivisions and Projections of Polytopes Jörg Rambau Dissertation (Advisor: Günter M. Ziegler, TU Berlin)

   
Topology

For details, we refer to the book ``Topology and Geometry'' by BREDON [17].

Definition 4.2.1   A   topological space is a pair $(X, \mathcal{O})$ where $X$ is a set and $\mathcal{O}$ is the   topology on $X$, that is, a family of subsets of $X$, called   open sets, such that

(i)

the empty set and $X$ are open,

(ii)

the intersection of two open sets is open,

(iii)

the union of any collection of open sets is open.

If the topology on $X$ is fixed, we often denote the topological space $(X, \mathcal{O})$ by $X$.

A subfamily $\mathcal{B}$ of open sets is a   basis of $\mathcal{O}$ if every open set is the union of sets in $\mathcal{B}$. It is a   subbasis if the set of all finite intersections of sets in $\mathcal{B}$ is a basis. In these cases we say that   $\mathcal{B}$ generates $\mathcal{O}$.

Let $(X, \mathcal{O})$ be a topological space and $x \in X$. A subset $N \subseteq X$ is a   neighborhood of $x$ if there is an open set $O \in \mathcal{O}$ with $x \in O$ and $O \subseteq N$. Let $(Y,\mathcal{U})$ be another topological space. A function $f: X \to Y$ is   continuous at $x$ if for any neighborhood $N$ of $f(x)$ the set $f^{-1}(N)$ is a neighborhood of $x$. A function $f$ is   continuous if it is continuous at every $x \in X$. A   homeomorphism is a bijective continuous map whose inverse is continuous as well.

What we really need is the standard topology of metric spaces, as on the Euclidean space $\mathbb{R} ^d$.

Definition 4.2.2   A   metric space is a pair $(X, d)$ where $X$ is a set and $d$ is a map from $X \times X$ to $\mathbb{R}$ such that

(i)

$d(x,y) \ge 0$, and $d(x,y) = 0$ if and only if $x = y$,

(ii)

$d(x,y) = d(y,x)$,

(iii)

$d(x,z) \le d(x,y) + d(y,z)$.

If $d(X) := \sup_{x,y \in X}d(x, y)$ exists then it is called the   diameter of $X$. If the metric on $X$ is fixed, we often denote the metric space $(X, d)$ by $X$.

The   topology induced by $d$ on $X$ is generated by the   open balls $B(x_0,r) := \setof{x \in X}{d(x, x_0) < r}$ for $x_0 \in X$ and $r \in \mathbb{R}$.

From now on all spaces are assumed to be metric spaces.

Lemma 4.2.3   Let $(X, d)$ and $(X',d')$ be metric spaces. Then $f: X \to X'$ is continuous if and only if for all $\epsilon > 0$ and all $x \in X$ there is a $\delta > 0$ such that $d'(f(x), f(y)) < \epsilon$ for all $y \in X$ with $d(x, y) < \delta$.

Corollary 4.2.4   If for $f: (X, d) \to (X', d')$ there is a constant $c$ with

$$d'(f(x),f(y)) \le c \cdot d(x,y) \quad \text{for all $x,y \in X$ },$$

then $f$ is continuous.

Lemma 4.2.5   The   product

$$(X,d) \times (X',d') := (X \times X', \max (d, d'))$$

is a metric space.

The following example is important in Section 2.3.

Example 4.2.6   The $1$-dimensional standard sphere

$$S^1 := \setof{z \in \mathbb{C} }{\norm{z} = 1}$$

is a metric space with the induced metric of  $\mathbb{C}$. Its diameter is $2$.

The product $S^1 \times \dots \times S^1$ is a metric space with the maximum metric

$$d((z_1, \dots , z_k), (z_1', \dots , z_k')) := \max (\norm{z_1' - z_1}, \dots , \norm{z_k' - z_k}).$$

This gives again a diameter of $2$.

Definition 4.2.7   Let $f,g: X \to Y$ be continuous. A   homotopy from $f$ to $g$ is a continuous (in both coordinates) map

$$H: \left\{ \begin{array}{rcl} X \times [0,1] & \to & Y,\\ (x,t) & \mapsto & H(x,t), \end{array} \right.$$

such that

$$H(x,0) = f(x) \quad \text{and} \quad H(x,1) = g(x).$$

Let $A \subseteq X$ be a subspace of $X$. If $H$ is a homotopy from $f$ to $g$ with $H(x,t) = f(x) = g(x)$ for all $x \in A$ then $H$ is a homotopy   relative $A$. The maps $f$ and $g$ are   homotopic (relative $A$) if there exists a homotopy from $f$ to $g$ (relative $A$). A continuous map $f: X \to Y$ is a   homotopy equivalence if there exists a function $g: Y \to X$ such that $g \circ f: X \to X$ is homotopic to the identity $\id_X$ on $X$, and $f \circ g: Y \to Y$ is homotopic to the identity $\id_Y$ on $Y$. In the case that such a map exists, $X$ and $Y$ are   homotopy equivalent. If $X$ is homotopy equivalent to a point then $X$ is   contractible.

Definition 4.2.8   A   path in $X$ from $x_0$ to $x_1$ is a continuous function $w: [0,1] \to X$ with $x_0 = w(0)$ and $x_1 = w(1)$. A path $w$ in $X$ is   null-homotopic if it is homotopic to a constant function $x_0: [0,1] \to X$. Two paths $w, v$ in $X$ are homotopic   relative $\partial [0, 1]$ if $w$ is homotopic to $v$ by a homotopy relative $\{ 0,1 \} \subset [0,1]$. In particular $v(0) = w(0)$ and $v(1) = w(1)$. For two paths $w, v: [0,1] \to X$ in $X$ with $w(1) = v(0)$, the   concatenation of $w$ and $v$ is the path

$$w \cdot v: \left\{ \begin{array}{rcl} [0,1] & \to & X,\\ ... ...{2} \le t \le 1$ }. \end{array} \right. \end{array} \right.$$

If $w(0) = w(1)$ then $w$ is a   closed path. Given a continuous function $f: X \to Y$ and a path $w$ in $X$ we define the path $f_* (w) := f \circ w: [0,1] \to Y$ in $Y$.

$X$ is   path-connected if for any two points $x$ and $y$ in $X$ there is a path in $X$ from $x$ to $y$. It is   $1$-connected if it is path-connected, and all paths are null-homotopic.

Lemma 4.2.9   If $w$ and $v$ are paths homotopic relative $\partial [0, 1]$ in $X$ and $f: X \to Y$ is continuous then the paths $f_*(w)$ and $f_*(v)$ are homotopic relative $\partial [0, 1]$ in $Y$.

Definition 4.2.10   Let $X$ be a topological space. A continuous function $p$ from a topological space $\Tilde{X}$ onto $X$ is a   covering of $X$ if for all $x \in X$ there is an open set $O_x \subset X$ with $x \in O_x$ such that

(i)

the set $p^{-1}(O_x)$ is the disjoint union of finitely many open sets $\Tilde{O}^{(i)}_x$ in $\Tilde{X}$, where $i = 1, \dots , r$,

(ii)

the restricted projection $p\vert _{\Tilde{O}^{(i)}_x}: \Tilde{O}^{(i)}_x \to O_x$ is a homeomorphism for all $i = 1, \dots , r$.

For a path $w$ in $X$ and a covering $p: \Tilde{X} \to X$, the   lifting of $w$ with starting point $\Tilde{x}_0 \in p^{-1}(w(0))$ is the unique path $L_p (w, \Tilde{x}_0) : [0,1] \to \Tilde{X}$ with $L_p (w, \Tilde{x}_0) (0) = \Tilde{x}_0$ and $p_* (L_p (w, \Tilde{x}_0)) = w$.

The   universal covering of $X$ is a covering $p: \Tilde{X} \to X$ where $\Tilde{X}$ is $1$-connected.

Example 4.2.11   The exponential function

$$\exp : \left\{ \begin{array}{rcl} \mathbb{R} & \to & S^1,\\ t & \mapsto & \exp(2\pi i t), \end{array} \right.$$

describes the universal covering of the standard $1$-sphere $S^1 \subset \mathbb{C}$, the set of all complex numbers with absolute value $1$.

Theorem 4.2.12 (Lifting Theorem)   Let $p: \Tilde{X} \to X$ be a covering, and let $w, v$ be paths homotopic relative $\partial [0, 1]$. Then $L_p (w, \Tilde{x}_0)$ and $L_p (v, \Tilde{x}_0)$ are homotopic relative $\partial [0, 1]$ for all $\Tilde{x}_0 \in p^{-1}(w(0))$.