= Solution
An <edge loop> is a finite sequence of vertices $v_0,\ldots,v_m$, starting and ending at $b$, in which consecutive distinct vertices span an edge. Two such sequences are equivalent when related by finitely many elementary edge homotopies, with endpoints fixed: insert or delete a repeated vertex, cancel or insert a spur $(u,v,u)$, and replace $(u,v,w)$ by $(u,w)$ or conversely when the vertices lie in one <simplex>. Repetitions are allowed in this last convention; a nondegenerate move uses a triangle.
The <edge-loop group> $E(K,b)$ consists of these equivalence classes, with product given by concatenating the sequences at $b$. Elementary moves remain valid inside a longer concatenation, so this product is well defined. Concatenation is associative after the harmless repeated base vertex is removed; the constant <edge loop> is the identity. Reversing a sequence gives its inverse, because a sequence followed by its reverse cancels by successive spurs. Thus these operations do form a group.
Realize an <edge loop> as a piecewise linear path in the <geometric realization> $|K|$. A spur contracts in an edge, and a triangle move is a <homotopy> across that triangle, always fixing endpoints. Hence realization gives a well-defined map $\theta$ to the <fundamental group>. Realization respects concatenation up to endpoint-preserving reparametrization, so $\theta$ is a <homomorphism>.
The relative <Simplicial approximation theorem> says that a continuous map from a finite complex to a <geometric realization> is homotopic, after sufficiently fine subdivision of its domain, to a simplicial map, with a chosen already simplicial subcomplex kept fixed. Apply its based-<circle> version to a loop, keeping its marked vertex at $b$. The resulting subdivided simplicial <circle> traverses edges, so every <based homotopy> class has an edge-loop representative. Thus \b[$\theta$ is surjective].
For the full two-skeleton, use the triangles through $b$ to replace each edge $(u,v)$ of a loop by $(u,b,v)$. Degenerate cases use repeated-vertex or spur moves. The resulting path contracts by spurs, so its <edge-loop group>, and therefore its <fundamental group>, is trivial. This also covers one or two vertices, when only repeated vertices and spurs are needed.
If the deleted triangle has vertices $u,v,w$, choose a fourth vertex $t$. Its boundary is filled by the three present triangles $(t,u,v)$, $(t,v,w)$ and $(t,w,u)$: their union is the cone from $t$ on that boundary. Every elementary <homotopy> that formerly crossed the missing triangle can therefore be replaced by a <homotopy> through these three triangles, staying in $|L|$. Since the one-skeleton has not changed, the same contraction of any <edge loop> is available in $L$. Surjectivity of its realization map gives
$$
\boxed{\pi_1(|K|,b)=1,\qquad\pi_1(|L|,b)=1\quad(n\geq4\text{ for }L)}.
$$
Back to article page
