Past exam of the topology course of the University of Oxford / 2015 / 3 / a

The standard simplex of dimension $n$ isIt is the convex hull of the coordinate vectors, and its faces of a simplex arise by setting selected coordinates to zero.

An abstract simplicial complex is a pair $(V,\Sigma )$ where $V$ is the vertex set and $\Sigma$ consists of finite nonempty subsets of $V$, contains every singleton, and contains every nonempty subset of each member. A member with $n+1$ vertices is an $n$-simplex.

Its geometric realization is obtained by gluing one standard simplex for each member along the faces specified by common vertex subsets. Equivalently, its points have barycentric coordinates $(t_v{)}_{v\in V}$ with finite support, nonnegative entries summing to one, and support a simplex of $K$. Use the topology induced by this gluing, equivalently the weak topology of a simplicial realization.

List the finite vertex set as $V=\{v_1,\dots ,v_N\}$. Map a point to its barycentric coordinates:On each simplex this is an affine map and therefore continuous. The definitions agree on every common face, so the quotient topology defining the realization makes $f$ continuous on all of $|K|$. Barycentric coordinates are unique after zero coordinates are added, so $f$ is injective.

Since Euclidean space is Hausdorff, distinct images have disjoint open neighbourhoods. Their inverse images separate the original points. Hence $|K|$ is Hausdorff. A continuous injection into a Hausdorff space is enough for this conclusion; no prior Hausdorff assumption on the domain is being used.

An $n$-dimensional topological manifold is a Hausdorff, second countable topological space in which every point has an open neighbourhood homeomorphic to an open subset of ${\mathbb{R}}^n$. Equivalently, charts can be chosen onto open balls. This is the without-boundary convention.

A topological surface is a two-dimensional topological manifold. When connectedness is included in the convention for a surface, impose it as well; in the classification statement below the surface is explicitly connected. A closed surface means a compact surface without boundary, rather than merely a closed subset of another space.

Here is an explicit simplicial complex on six vertices. Take every subset of one of the following ten triangular facets:For example, $135$ denotes $\{1,3,5\}$, with all its edges and vertices included. Its one-skeleton has all fifteen edges of the complete graph. Every edge belongs to exactly two triangles. The links of a simplex at the six vertices are the following cycles, with the first vertex understood to repeat at the end:Thus every vertex has a disk neighbourhood, as do edge and face interior points. The realization is a compact connected surface without boundary. Its Euler characteristic isThe classification theorem for surfaces identifies the unique closed connected surface with this Euler characteristic as the real projective plane. Consequently this complex triangulates ${\mathbb{RP}}^2$. The explicit facet and link checks make this a genuine simplicial complex, not a polygon whose boundary identifications have been mistaken for simplicial faces.