Singular Simplices

Algebraic Topology · Axiom Academy

Building blocks for singular homology theory The standard n-simplex is the fundamental building block. It is the convex hull of n+1 vertices in Euclidean space. For n=0, we get a point. For n=1, a line segment. For n=2, a triangle. For n=3, a tetrahedron. A singular n-simplex in a topological space X is a continuous map from the standard n-simplex to X. The key insight: we can use these maps to systematically explore the structure of any topological space, regardless of whether it has a natural triangulation. Singular n-chains are formal linear combinations of singular n-simplices with integer coefficients.

This is the written version of the interactive lesson above. See the full Algebraic Topology course.