The <universal cover> $\widetilde X$ is simply connected. Hence the closed combinatorial path $\gamma\sigma^{-1}$ is <null-homotopic> in its two-dimensional <CW complex>. The <van Kampen lemma> supplies a finite <disc diagram> mapping to $\widetilde X$ with precisely that boundary path.

Choose such a diagram with the smallest possible number of two-cells, and then with the smallest number of edges. If two adjacent cells formed a cancellable pair, removing that pair would preserve the boundary path and reduce the area, a contradiction. Thus
$$
\boxed{D_\sigma\to\widetilde X\text{ can be chosen reduced}.}
$$
This does not assume that $\widetilde Y$ is simply connected. Only the loop's null-homotopy in $\widetilde X$ is used. If the two boundary paths agree, a diagram with no two-cells is allowed; if they share initial or final segments, the diagram may have tree portions or cut vertices. A <disc diagram> need not be an embedded topological disk.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 133 / 2 / c / Solution

{c}

A closed combinatorial path in a <2-complex> is <null-homotopic> exactly when it is the boundary path of a <disc diagram> over that complex. A finite cellular null-homotopy can be arranged into such a planar diagram; conversely the contractible diagram supplies a null-homotopy. In a <group presentation>, this is the geometric form of expressing a trivial word as a product of conjugates of <relators>.

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 Incoming links: van Kampen lemma