Problem 1
It records n-dimensional cycles modulo those that bound higher-dimensional chains.
Faculty solution guide
Homology Groups and Dimensional Holes
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Homology Groups and Dimensional Holes.
Problem 4
The argument must address forming cycles modulo boundaries and computing basic examples with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in topological data analysis, image shape, mesh processing, sensor networks, and manifold invariants, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.