Problem 1
Paths lift uniquely after choosing a starting point, and homotopies lift under compatible initial data.
Faculty solution guide
Covering Spaces and Lifting
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 Covering Spaces and Lifting.
Problem 4
The argument must address using unique path and homotopy lifting to analyze maps with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in robot motion, periodic systems, knot complements, phase spaces, and obstruction theory, 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.