Questions: Homology and Cohomology

Question 1

A chain complex C_* satisfies H_n(C) = 0 for all n. What does this mean about the complex?

• A. All chain groups C_n are zero — the complex is trivially empty
• B. The complex is exact at every degree — every cycle is a boundary
• C. All boundary maps d_n are zero maps, so nothing maps anywhere
• D. The complex has only even-degree terms; odd-degree terms are absent

Answer: (B)

H_n(C) = ker(d_n)/im(d_{n+1}) = 0 means ker(d_n) = im(d_{n+1}) at each degree — every cycle is a boundary. This is exactness. The chain groups C_n themselves can be very large and non-trivial; zero homology says nothing about the size of the groups, only about the relationship between consecutive maps. This is a common misconception: zero homology does not mean the complex is empty or trivial, but that it has the 'right' map structure — a strong and often useful condition.

Question 2

A topologist wants to distinguish CP² (complex projective plane) from S² ∨ S⁴ (the wedge sum). Both spaces have identical homology groups in every degree. Which algebraic structure detects the difference?

• A. The kernel of the boundary map in degree 2
• B. The cup product structure in cohomology
• C. The connecting homomorphism in the long exact sequence
• D. The torsion subgroups of the chain groups

Answer: (B)

H*(CP²; ℤ) and H*(S² ∨ S⁴; ℤ) are isomorphic as graded abelian groups — homology cannot distinguish them. But in cohomology, the generator α ∈ H²(CP²) satisfies α ∪ α ≠ 0 (the cup product generates H⁴), while on S² ∨ S⁴ any cup product of positive-degree classes is zero. The cup product is a ring structure on cohomology with no homology analogue, demonstrating that cohomology sometimes carries strictly more information.

Question 3

If the homology group H_n(C) = 0, then the chain complex is expected to be trivial — most of the chain groups C_n are zero.

• T. True
• F. False

Answer: False

H_n(C) = 0 means the complex is exact at degree n (ker d_n = im d_{n+1}), which is a condition on the relationship between maps, not on the sizes of the groups. A non-trivial example: the complex 0 → ℤ →×2 ℤ → ℤ/2 → 0 is exact (has zero homology) but involves non-trivial groups throughout. Exactness is a structural property; zero homology is a powerful condition that says cycles and boundaries coincide perfectly at each degree.

Question 4

Cohomology can distinguish spaces that homology cannot, because the cup product in cohomology carries information about how cohomology classes intersect that homology groups alone do not capture.

• T. True
• F. False

Answer: True

As illustrated by CP² versus S² ∨ S⁴, cohomology rings (equipped with the cup product) detect multiplicative structure invisible to homology groups. The cup product α ∪ β ∈ H^{p+q} encodes intersection data between p- and q-dimensional classes. This extra structure makes cohomology essential in algebraic topology, algebraic geometry (de Rham, sheaf, and étale cohomology), and physics (characteristic classes, anomaly cancellation). Cohomology is dual to homology but is not equivalent to it in general.

Question 5

What does the homology group H_n(C) actually measure, and why is the definition H_n = ker(d_n) / im(d_{n+1}) geometrically meaningful?

Model answer: H_n(C) measures the failure of the chain complex to be exact at degree n — it counts n-cycles (closed chains with no boundary, in ker d_n) that are not themselves boundaries of (n+1)-chains (not in im d_{n+1}). In topology, these are genuine 'holes': loops not bounding disks, voids not enclosing solid regions. H_0 counts connected components, H_1 counts 1-dimensional holes (loops), H_2 counts enclosed voids, and so on. The quotient is meaningful because d² = 0 guarantees im d_{n+1} ⊆ ker d_n — without this, the quotient wouldn't even be well-defined.

The condition d² = 0 is the algebraic foundation that makes homology possible. It means every boundary is a cycle, so cycles form a group containing boundaries as a subgroup, and we can take the quotient. The size of H_n reflects how many independent non-boundary cycles exist — the 'topological complexity' at dimension n.