Problem 1
For a finite complex, χ equals the alternating sum of cell counts and also the alternating sum of Betti numbers.
Faculty solution guide
Euler Characteristic and Homological Counting
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 Euler Characteristic and Homological Counting.
Problem 4
The argument must address proving alternating cell counts agree with alternating Betti numbers 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.