lecture notes: 10.12.12

Topic: polygons, polyhedra, cell complexes and data structure for these.

$n$-manifolds looks locally like $\mathbb{E}$. Further a $n$-manifold with boundary has points with neighboorhoods like a halfspace $\{x\in \mathbb{E}^n | x_n\geq 0\}$. Compact manifolds (with boundary) are the ones that can be represented in the computer, say by a finite triangulation.

Definition: A closed $n$-manifold is a compact manifold without boundary.

Each component of a $0$-manifold is a point. The manifold is compact iff it is finite. A connected $1$-manifold is $\bigcirc\cong\mathbb{S}^1, \:\mid\:\cong [0,1],\:\updownarrow \:\cong\mathbb{R}\cong(0,1)$ or $\uparrow \:\cong\mathbb{R}_{\geq 0} \cong [0,1)$. Thus, a compact $1$-manifold has finitely many components, each is $\mathbb{S}$ or $\mathrm{I}$.
Recall: each component of a compact $2$-manifold is $\sum_{g,k}$ if orientable or $\mathrm{N}_{h,k}$ if not.
Represent a compact $1$-manifold by gluing edges together: Start with n edges and some identifications of the end points (an equivalence relation on the set of $2n$ end points). In general this gluing gives a multigraph with $n$ edges and less than $2n$ vertices. The size of each equivalence class is the degree of the resulting vertex. A finite multigraph can be realized this way iff there is no vertex of degree $0$.

Lemma: A multigraph is a $1$-manifold iff every vertex has degree $1$ or $2$.

Definition: The body is exactly the set of all vertices of degree $1$.

For $2$-dimensional manifolds (and cell complexes) we start with a finite collections of polygons and a rule for gluing edges. Equivalently, we can require that all the polygons are triangles.

Definition: An oriented $n$-gon is topologically an oriented closed disk with n marked boundary-points $a_i$ in cyclic order.

The gluing rule is an equivalence relation on the set of all oriented edges with the porperty that $e\sim f \Leftrightarrow -e \sim -f$. The equivalence classes are the oriented edges of the resulting $2$-complex.
The degree or valence of the resulting edfe is the cardinality of the equivalence class. The resulting compact space is a $2$-manifold iff all edges has valence $1$ or $2$ and edges of valence $1$ form the boundary. We get a manifold if we start with pairing of the oriented edges. The manifold is closed (no boundary) if all edges get paired.

The $2$-complex we get is always purely $2$-dimensional (no edges of valence $0$, no isolated vertices). Not every purely $2$-dimensional complex can be obtained by our construction. We’ve given a “top-down” $\:$construction of complexes. Start with to-dimensional cells gluing their face.
The other construction of cell complexes is “bottom up”.$\:$Start with a finite set of vertices. Then sew in a finite collection of edges (to head and tail vertices). Finally sew in $n$-gons by attaching boundary to a cycle of $n$ oriented edges.
With this second construction edge valence is $0$ or $2$ is necessary but not sufficient to get a manifold with boundary. We need to test that the link of each vertex is connected. Assuming the valence conditions, the link of each vertex is a $1$-manifold – we just need to check if it is $\mathrm{I}$ or $\mathbb{S}^1$.
Usually we have geometric information about how a complex is sitting in some ambient space. E.g. each vertex has coordinates. Usually edges are straight lines in the ambient space.
We may need to mark edges to say which “way around”$\:$ in the ambient space they go. If the ambient space is $\mathbb{R}^3/\:\mathrm{G}$ for some crystallographic group $\mathrm{G}$, then we often represent vertices as points $v\in\mathbb{R}^3$. An edge from $v$ to $w$ is marked with $g\in\mathrm{G}$ to show that it is the straight line $v$, $g\cdot w$ is one of the preimages.
Sometimes multiple edges are usefull even if the start in the same position.

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