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Two based edge loops are equivalent under insertion or deletion of stationary steps, cancellation or insertion of a backtracking spur, and replacement of two sides of a triangle by its third side. The edge-loop group consists of the resulting classes, with concatenation as product and reversal as inverse. Each elementary move can be performed inside a longer concatenation, so multiplication is well defined. Associativity follows from concatenation, the stationary loop is the identity, and a loop followed by its reversal cancels by spurs.

Realization induces a homomorphism to the fundamental group of the geometric realization, because each move is a homotopy in an edge or a triangle. The relative Simplicial approximation theorem applied to based circles makes this map surjective. The full edge-path theorem additionally proves injectivity by approximating homotopies; that stronger assertion is not needed for the group construction or the surjectivity argument.

The Simplicial approximation theorem states that a continuous map between realizations of finite simplicial complexes becomes homotopic to a simplicial map after sufficiently many barycentric subdivisions of the domain. More precisely, a vertex assignment is a simplicial approximation when the image of each open star is contained in the open star of its assigned vertex; sufficiently fine subdivision permits such an assignment. The resulting straight-line homotopy stays in the appropriate target simplices.

The vertices of a barycentric subdivision are the barycentres of nonempty faces. An $n$-simplex has $2^{n+1}-1$ such faces. A top-dimensional subdivided simplex corresponds to a full flag of faces of sizes $1,2,\dots ,n+1$, equivalently to an ordering of the original vertices. ThusTriangulate both copies of the sphere using finite complexes. For each fixed subdivision level there are only finitely many vertex maps to the finite target vertex set, hence only finitely many simplicial maps. Every continuous map is homotopic to one of them at some finite level. A countable union of finite sets is countable, proving at most countably many homotopy classes without needing their classification by degree.

A simplicial approximation to $f:|K|\to |L|$ is a simplicial map $g:K\to L$ satisfying the open-star conditionfor every vertex $v$. Equivalently, whenever $f(x)$ lies in the interior of a simplex of $L$, $|g|(x)$ lies in that simplex. In particular $f$ and $|g|$ are joined by the simplexwise straight-line homotopy.

The Simplicial approximation theorem says that, for finite $K$, every continuous $f$ has such an approximation after sufficiently many barycentric subdivisions of $K$. For general complexes one uses a subdivision adapted to the open-star cover; a single uniform number of barycentric subdivisions is the compact-domain version.

Given triangulations $X\cong |K|$, $Y\cong |L|$, choose a simplicial approximation $g$ on a suitable subdivision $K^{\prime }$. Its chain map sends an oriented simplex $[v_0,\dots ,v_r]$ to $[g(v_0),\dots ,g(v_r)]$, with repeated-vertex simplices sent to zero. It commutes with the boundary operator and induces $g_{\ast }$. Compose this with the subdivision isomorphism $H_r(K)\to H_r(K^{\prime })$ and the triangulation identifications to define $f_{\ast }:H_r(X)\to H_r(Y)$.

For $z\mapsto z^n$, a positively oriented fundamental loop winds $n$ times around the target circle. With integer coefficients,All higher homology groups vanish. The $H_1$ result can also be read from a subdivision into $n$ copies of a target fundamental cycle.

The Simplicial approximation theorem states that if $K$ is a finite simplicial complex, $L$ is a simplicial complex, and $f:|K|\to |L|$ is continuous, then for some $r\ge 0$ there is a simplicial approximation $g:K^{(r)}\to L$ to $f$.

The target open stars form an open cover of $|L|$, so their inverse images under $f$ form an open cover of the compact metric space $|K|$. The Lebesgue number lemma supplies $\delta >0$ such that every subset of diameter below $\delta$ lies in one inverse image. Choose $r$ so large that every vertex star in $K^{(r)}$ has diameter below $\delta$; this is possible because its diameter is at most twice $\mu (K^{(r)})$ and the mesh tends to zero.

For each vertex $v$ choose a vertex $w(v)$ of $L$ such thatIf $v_0,\dots ,v_q$ span a simplex, a point in its relative interior belongs to all their open stars. Its image belongs to all the open stars of $w(v_0),\dots ,w(v_q)$, so these target vertices span a simplex. Thus $v\mapsto w(v)$ extends to a simplicial map $g$, and its defining inclusion says exactly that $g$ approximates $f$. This proves the theorem.

Triangulate $S^n$ and $S^m$. By the Simplicial approximation theorem, after subdividing the domain, a continuous map $f:S^n\to S^m$ is homotopic to the realization of a simplicial map $g$. Since $n<m$, the image of $|g|$ lies in the $n$-skeleton of the target triangulation and therefore misses an interior point of every $m$-simplex. A sphere with one point removed is homeomorphic to the Euclidean space ${\mathbb{R}}^m$, which is contractible. Hence $|g|$, and therefore $f$, is homotopic to a constant map. Thus

For a based space $(X,x_0)$, the homotopy groupis the set of based homotopy classes, with its usual concatenation operation. A map $f:X\to Y$ is a weak homotopy equivalence when it induces a bijection on path components and an isomorphismfor every $k\ge 1$ and every basepoint $x$. It is an n-connected map when it is bijective on ${\pi }_k$ for $k<n$ and surjective on ${\pi }_n$; equivalently, every homotopy fiber is $(n-1)$-connected.

A CW complex is built from a discrete set of zero-cells by successively attaching $k$-discs along maps from their boundary spheres, with the weak topology and closure-finiteness conditions. Its filtration by skeleta is the CW filtration.

For any space $X$, form its singular simplicial set $\mathrm{Sing}X$. Its geometric realization of a simplicial set is a CW complex, with one cell for each nondegenerate singular simplex, and evaluation givesThe Simplicial approximation theorem identifies based maps and homotopies from finite simplicial spheres into $|\mathrm{Sing}X|$ with singular simplices in $X$. Consequently $ϵ$ induces a bijection on components and isomorphisms on all homotopy groups. Thus every space admits a CW approximation.

The vanishing assumptions do not permit removal of all $n$-cells. Take $n=2$ andThen ${\pi }_2(X)=0$, and homology of a finite cyclic group gives $H_2(X;\mathbb{Z})=0$. If a connected CW complex $Y$ had no two-cells, attaching cells of dimension at least three would not change the fundamental group of its one-skeleton. Hence ${\pi }_1(Y)$ would be a free group. A weak equivalence $Y\to X$ would instead give ${\pi }_1(Y)\cong \mathbb{Z}/r$, which is nontrivial and finite and therefore not free. No such $Y$ exists.

An edge loop is a finite sequence of vertices $v_0,\dots ,v_m$, 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 $\theta$ 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

A map satisfying this condition is the object whose existence after subdivision is guaranteed by the Simplicial approximation theorem.

A simplicial map $g:K\to L$ is a simplicial approximation to a continuous map $f:|K|\to |L|$ whenfor every vertex $v$ of $K$.