Questions: Simplicial Complexes

Question 1

Which of the following is NOT a valid simplicial complex?

• A. Two triangles sharing a common edge
• B. A triangle with one of its edges removed
• C. Three edges forming a triangle (without the filled interior)
• D. A single vertex

Answer: (B)

A simplicial complex must contain every face of every simplex it contains. A triangle (2-simplex) has three edges and three vertices as faces. Removing one edge while keeping the triangle violates the face condition. The other options are all valid: two triangles sharing an edge satisfy all face conditions, the boundary of a triangle (three edges and three vertices) is a valid 1-dimensional complex, and a single vertex is the simplest possible complex.

Question 2

A simplicial complex K has 4 vertices, 6 edges, and 4 triangular faces. What familiar space does the geometric realization |K| resemble?

• A. The 2-sphere S²
• B. The torus T²
• C. The real projective plane RP²
• D. The 2-disk D²

Answer: (A)

The Euler characteristic is V - E + F = 4 - 6 + 4 = 2, which matches S². This is a minimal triangulation of the 2-sphere — the boundary of a tetrahedron. The torus has Euler characteristic 0, RP² has Euler characteristic 1, and D² has Euler characteristic 1. While Euler characteristic alone does not uniquely determine a surface, a complex with 4 vertices, 6 edges, and 4 faces where every edge is shared by exactly two triangles is the boundary of the tetrahedron.

Question 3

Every topological space can be given the structure of a simplicial complex.

• T. True
• F. False

Answer: False

Not every topological space is triangulable. A simplicial complex has a very rigid combinatorial structure, and its geometric realization is always a nice space (a polyhedron). Spaces that are not locally Euclidean, or certain exotic manifolds, cannot be triangulated. Even among manifolds, there exist topological manifolds in dimension 4 and higher that admit no triangulation (Manolescu, 2013). However, for the purposes of algebraic topology, many important spaces — surfaces, CW complexes, smooth manifolds — can be triangulated or approximated by simplicial complexes.

Question 4

An abstract simplicial complex on vertex set {a, b, c, d} contains the 2-simplices {a,b,c} and {a,b,d}. What other simplices must it contain?

Model answer: All faces of both 2-simplices: the 1-simplices {a,b}, {a,c}, {b,c}, {a,d}, {b,d}, and the 0-simplices {a}, {b}, {c}, {d}.

The defining property of a simplicial complex is that every face of every simplex in the complex must also be in the complex. A 2-simplex {a,b,c} has faces {a,b}, {a,c}, {b,c}, {a}, {b}, {c}, and the empty set. Similarly for {a,b,d}. The complex need NOT contain {c,d} or {b,c,d} or {a,c,d} — those are not faces of either given simplex. Note that {a,b} is shared by both 2-simplices, which is allowed; two simplices may intersect along a common face.

Question 5

Why are simplicial complexes useful in algebraic topology, given that they impose a rigid combinatorial structure on spaces?

Model answer: Simplicial complexes reduce topology to combinatorics: computing homology groups becomes a matter of linear algebra over the chain groups generated by simplices. The boundary maps are determined entirely by the combinatorial data (which simplices are faces of which), making homology algorithmically computable. Any two triangulations of the same space yield isomorphic homology groups, so the combinatorial rigidity does not affect the topological invariants we extract.

The key insight is that we trade topological generality for computational tractability. The rigidity of simplicial complexes is a feature, not a bug: it lets us define boundary operators as explicit matrices and compute homology via linear algebra. The price is that some spaces need very many simplices to triangulate — which motivates the development of singular homology (which works for arbitrary spaces) and CW complexes (which are more economical).