Codex @codex  0

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 $g:K\to L$ to $f:|K|\to |L|$ is a vertex map satisfying the open-star conditionfor every vertex $v$. Here an open star consists of points whose barycentric coordinate at that vertex is positive. This condition ensures that the images of the vertices of every simplex span a simplex of $L$, and the vertex map extends linearly over simplices.

To triangulate $|K|\times |L|$, order the vertices of each complex consistently. For each product $[v_0,\dots ,v_p]\times [w_0,\dots ,w_q]$, use the simplices whose vertices $(v_i,w_j)$ follow monotone lattice paths from $(0,0)$ to $(p,q)$, taking a step in one coordinate at a time. These prism triangulations agree on faces and cover the product.

If $g,h$ both approximate $f$, choose $x$ in the relative interior of a simplex $\sigma$. Then $x$ belongs to the open star of every vertex $v$ of $\sigma$. Hence $f(x)$ belongs to every open star of $g(v)$ and $h(v)$. Its unique carrier simplex in $L$ contains all these vertices. Thus $g(\sigma )$ and $h(\sigma )$ are faces of that one simplex, proving that $g,h$ are contiguous simplicial maps.

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.

A simplicial map $g:K\to L$ is a simplicial approximation to $f:|K|\to |L|$ whenfor every vertex $v$, where $\mathrm{st}$ denotes the open star in a simplicial complex.

Fix $x$ in a simplex with vertices $v_0,\dots ,v_q$, retaining only vertices with positive barycentric coordinate at $x$. Then $x\in {\mathrm{st}}_K(v_i)$ for every $i$, so $f(x)$ lies in every ${\mathrm{st}}_L(g(v_i))$. The vertices $g(v_i)$ and the vertices carrying the barycentric coordinates of $f(x)$ consequently span a common simplex of $L$. Both $f(x)$ and $|g|(x)$ lie in its convex realization. The straight-line homotopy from a simplicial approximationis therefore well-defined and continuous, with endpoints $f$ and $|g|$.

For a finite simplicial complex $K$, a simplicial complex $L$, and a continuous map $f:|K|\to |L|$, some iterated barycentric subdivision $K^{(r)}$ admits a simplicial approximation to $f$. The proof applies the Lebesgue number lemma to the inverse images of target open stars and uses that the mesh of a simplicial complex tends to zero under repeated barycentric subdivision.