Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 12G / c

False. The constant map to $E$ satisfies it: $E\in I$ and also $E\in J\subset J\cup K$.

True. At the vertices the side conditions forceOn each side, $g$ remains in that same side, so the boundary restriction is homotopic within the boundary to the identity and has degree one. If it extended continuously over the filled triangle with image in the boundary circle, that degree-one boundary map would be null-homotopic, which is impossible. Equivalently, a sufficiently fine simplicial approximation would contradict the parity form of Sperner's lemma proved in part (b).