Math 2023, 44: Simplicial Approximation

Short Chapter this week, but not everything has to be a long-form project. From an entry-level topology course, the readers should be familiar with both the fundamental group and its relation to homologies, from which the Borsuk-Ulam Theorem is usually derived. These form the fundamentals for the "simplicial approximation theorem". It's used to bring spaces into a structure of simplicial complexes, which, while not as widely used as CW complexes, will still introduce a structure to the space which is familiar. Occasionally the homology lying over spaces constructed from simplicial will be beneficial also. For simplicial complexes K and L, a map f: K → L is considered simplicial, if it sends each simplex of K to a simplex of L by associating vertices to vertices linearly. The Simplicial Approximation Theorem states that f is homotopic to a map that is simplicial with respect to some iterated barycentric subdivision of K (subdivision, since f need not be surjective). We note St ξ for the star of a simplex ξ and st ξ for its open star (i.e. an open set with the closure St ξ). For vertices in a simplicial complex, the intersections of their open stars is non-empty, if a simplex ζ exists which contains all vertices. The set of all these vertices then (trivially) create a simplex ξ in ζ, meaning that the intersection of all open stars of all vertices is exactly st ξ. The Simplicial Approximation Theorem can be proven by choosing a metric K restricting the standard Euclidean metric on each simplex of K, and finding a Lebesgue number of an open cover of K, so that each simplex in the subdivision of K is less than half this number in diameter. The closed star of each vertix has a diameter less than the Lebesgue number. Mapping this with a map f, as constructed will map each vertex v to some g(v) of L, which will define g as a map that extends to a simplicial map g: K → L. If one wants the maps to preserve basepoints, g can be set so that it maps vertices that are already sent from K to L by f to themselves.

A homomorphism Ψ: ℤⁿ → ℤⁿ with the matrix [aᵢⱼ] and the standard trace operator. Conjugate matrices have the same trace, so tr Ψ is independent of the chosen basis. If Ψ is a homomorphism of a finitely generated abelian group A, then tr Ψ is the trace of the induced homomorphism φ: A/Torsion→A/Torsion. Applying this to CW-complexes gives rise to a new quantity, the "Lefschetz number"

If f is the identity (or homotopic to it), then τ(f) is the Euler characteristic. From this emerges the Lefschetz fixed point theorem, stating that if X is a finite simplicial complex (or any retract of a finite simplicial complex), and f: X → X is a map with τ(f) ≠ 0, then f has a fixed point.

To see this, note that every compact, locally contractible space, for which an n exists, so that the space can be embedded into ℝⁿ, is a retract of a finite simplicial complex.

Having seen a method to approximate continuous maps into homotopic simplicial maps, a similar method for spaces should be possible. There are at least spaces, which are homotopy-equivalent to simplicial complexes. The most common class of spaces in algebraic topology are CW complexes, so it makes sense to start there. Every CW complex X is homotopy equivalent to a simplicial complex, chosen to be of the same dimension. The proof can be approached inductively, by showing the statement for unions of subcomplexes, homotopy equivalent to the skeleta of X. The inductive step will introduce an (n+1)-cell of X, attached by a map φ. A map corresponding to φ under homotopy equivalence is homotopic to a simplicial map by the simplicial approximation theorem. The union of the subcomplexes and the (n+1)-cell is homotopy equivalent to the skeleton of X and (n+1)-cell. It can be helpful to construct simplicial analogs of mapping cones to see the last step, instead of going through the mapping cone directly. It really depends on how comfortable one is with category theory.