Simplicial Complexes

Algebraic Topology · Axiom Academy

Building blocks for algebraic topology: constructing topological spaces from vertices, edges, and higher-dimensional simplices 1. Simplices: The Building Blocks A simplex is the generalization of a triangle to arbitrary dimensions. A 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a filled triangle, and a 3-simplex is a solid tetrahedron. A simplicial complex K is a collection of simplices satisfying two key properties: (1) every face of a simplex in K is also in K, and (2) the intersection of any two simplices in K is either empty or a common face of both. These conditions ensure that simplices "fit together nicely" without overlapping in unexpected ways. The geometric realization |K| of a simplicial complex K is the topological space obtained by actually gluing together all the simplices in K according to their face relations. This construction bridges the gap between combinatorial and topological perspectives, allowing us to compute topological invariants combinatorially. Understanding simplicial complexes requires working with concrete examples. The circle S 1 can be realized as the boundary of a 2-simplex (triangle), while the torus requires a more intricate construction.

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