Questions: Simplicial Homology Groups

Question 1

A simplicial complex K has H_0(K) ≅ Z ⊕ Z. What does this tell you about K?

• A. K has exactly two 1-dimensional holes
• B. K has exactly two connected components
• C. K is the 2-sphere
• D. K has two independent 0-cycles that do not bound any 1-chain

Answer: (B)

H_0 measures path-connectivity: its rank equals the number of connected components. Each connected component contributes one copy of Z to H_0. With H_0 ≅ Z ⊕ Z, K has exactly two connected components. Option D is technically true (it restates the algebra) but B is the geometric interpretation. The 0-cycles are formal sums of vertices, and two vertices are homologous (differ by a boundary) if and only if they are connected by a path of edges.

Question 2

For the triangulated torus T², the homology groups are H_0 ≅ Z, H_1 ≅ Z ⊕ Z, and H_2 ≅ Z. What does the generator of H_2 represent?

• A. A single triangle on the surface of the torus
• B. The fundamental cycle — the sum of all oriented 2-simplices forming the closed surface, which encloses a cavity but does not bound any 3-chain in the complex
• C. The product of the two generating loops of H_1
• D. The Euler characteristic of the torus

Answer: (B)

The generator of H_2(T²) is the fundamental cycle: the sum of all 2-simplices of the triangulation, oriented consistently. This 2-cycle has zero boundary (the surface is closed — every edge is shared by exactly two triangles with compatible orientations), but it is not the boundary of any 3-chain (there is no 3-dimensional 'filling' in the complex). This generator detects the cavity enclosed by the torus. The single copy of Z reflects that the torus is a connected, orientable, closed surface.

Question 3

Two different triangulations of the same topological space always yield isomorphic simplicial homology groups.

• T. True
• F. False

Answer: True

This is a deep theorem: simplicial homology is a topological invariant, not dependent on the choice of triangulation. The proof proceeds by showing that simplicial homology agrees with singular homology (which is defined without reference to any triangulation). Alternatively, one can show directly that subdivisions and simplicial approximations relate different triangulations without changing homology. This invariance is what makes homology useful — it extracts topological information that does not depend on the combinatorial presentation.

Question 4

A simplicial complex K has H_1(K) ≅ Z ⊕ Z/2Z. Describe what the Z and Z/2Z summands represent geometrically.

Model answer: The Z summand corresponds to a 1-cycle (loop) that is not a boundary and has infinite order — going around it any number of times never becomes a boundary. The Z/2Z summand corresponds to a 1-cycle that is not itself a boundary, but going around it twice IS a boundary. This torsion element reflects a non-orientability phenomenon: the complex contains something like a Mobius band where traversing the core circle twice produces a boundary.

Torsion in homology detects subtle topological features beyond simple holes. The real projective plane RP² has H_1 ≅ Z/2Z because its core loop, traversed once, does not bound a 2-chain, but traversed twice it does (the loop 'unwraps' on the double cover S²). Spaces with both free and torsion summands in homology combine 'hole-like' and 'non-orientability-like' features.

Question 5

Explain why im(d_{n+1}) is always a subgroup of ker(d_n), and why this inclusion is what makes the quotient H_n = ker(d_n)/im(d_{n+1}) well-defined.

Model answer: The property d_n ∘ d_{n+1} = 0 means that for any (n+1)-chain c, d_n(d_{n+1}(c)) = 0. So d_{n+1}(c) is in ker(d_n) for every c, which means im(d_{n+1}) ⊆ ker(d_n). Since im(d_{n+1}) is a subgroup of the abelian group ker(d_n), the quotient ker(d_n)/im(d_{n+1}) is a well-defined abelian group. This quotient identifies two cycles as 'the same' precisely when they differ by a boundary, which is the fundamental equivalence relation of homology.

If im(d_{n+1}) were not contained in ker(d_n), the quotient would be undefined. The condition d ∘ d = 0 is not just a technical convenience but the core structural property that makes homology theory possible. Every construction in homological algebra — singular homology, cellular homology, de Rham cohomology — rests on this same foundational property of chain complexes.