Questions: Singular Simplices and Singular Chains

Question 1

Why is the singular chain group C_n(X) typically uncountably generated, even for simple spaces like the circle S^1?

• A. Because the circle has uncountably many points
• B. Because there are uncountably many continuous maps from Delta^n to S^1, and each one is a separate generator
• C. Because we use real coefficients instead of integer coefficients
• D. Because singular chains include maps from simplices of all dimensions

Answer: (B)

A singular n-simplex is ANY continuous map sigma : Delta^n -> X. Even for n = 0, the 0-simplices are points of X, and S^1 has uncountably many points. For n = 1, there are uncountably many continuous paths in S^1, each a separate generator of C_1(S^1). This is in stark contrast to simplicial chains, where a finite triangulation gives finitely generated chain groups. Despite the enormous size of singular chain groups, the homology they produce is finitely generated for reasonable spaces — the quotient ker/im collapses the uncountable redundancy.

Question 2

A singular simplex sigma : Delta^2 -> X that maps all of Delta^2 to a single point is called a degenerate simplex. Its boundary d_2(sigma) is zero.

• T. True
• F. False

Answer: False

The boundary d_2(sigma) = sigma|[v_1,v_2] - sigma|[v_0,v_2] + sigma|[v_0,v_1], where each face restriction is a singular 1-simplex mapping to the same single point. These three degenerate 1-simplices are distinct as formal generators but they are all the same singular simplex (the constant map to that point). So d_2(sigma) = sigma_point - sigma_point + sigma_point = sigma_point, not zero. The boundary of a degenerate 2-simplex is a degenerate 1-simplex (with coefficient 1 if there are an odd number of faces, which there are for n = 2). Degenerate simplices contribute to chains but cancel out in homology.

Question 3

What is the key advantage of singular homology over simplicial homology?

• A. Singular homology is easier to compute by hand
• B. Singular homology is defined for any topological space without needing a triangulation or cell structure
• C. Singular homology detects more topological features
• D. Singular homology uses simpler boundary operators

Answer: (B)

The defining advantage of singular homology is its universality: it works for any topological space X, because we only need the notion of 'continuous map into X.' No triangulation, cell decomposition, or combinatorial structure is required. The price is that the chain groups are enormous (typically uncountably generated), making direct computation impractical. In practice, we compute singular homology indirectly using tools like the Mayer-Vietoris sequence, long exact sequences, and excision — or by showing it agrees with simplicial/cellular homology when those are available.

Question 4

Explain why the boundary formula d_n(sigma) = sum_{i=0}^{n} (-1)^i sigma ∘ F_i (where F_i is the i-th face inclusion Delta^{n-1} -> Delta^n) still satisfies d_{n-1} ∘ d_n = 0 in the singular setting.

Model answer: The proof is purely combinatorial and identical to the simplicial case. Applying d twice gives a double sum over pairs (i, j) where we omit the i-th vertex and then the j-th vertex. Each pair of omitted vertices appears twice — once as (i, j) with j < i and once as (j, i-1) with the shifted index — and with opposite signs due to the (-1)^i(-1)^j sign convention. These pairs cancel, giving zero. The proof depends only on the alternating sign formula and the combinatorics of face inclusions, not on any property of sigma or X.

This is a crucial observation: the chain complex structure (and therefore the well-definedness of homology) is a formal consequence of the alternating sum formula, independent of the nature of the simplices. The same identity works for simplicial, singular, and cellular boundary operators, which is why all these homology theories share the same foundational algebraic structure.