Video – book

Construction procedure

1. Take Disjoint union of Triangles (or higher simplices)m with order given on vertices.
2. Glue edges (or vertices) together by order-preserving maps.

Definition

A $\Delta$-complex is a space $X$ together with maps $e_\alpha^n: \Delta^n \to X$ ($\Delta^n$ the $n$-dimensional Simplex) such that

1. $\bigsqcup\limits_\alpha e_\alpha^n |_{\Delta^{\circ n}}: \bigsqcup\limits_\alpha \Delta^{\circ n} \to X$ is bijective. ( $\circ$ above means interior of the set (this is done to avoid having the boundaries where they overlap counted multiple times))
2. Every $e_\alpha^n |_{\text{face}_i}$ is some $e_{\beta(\alpha,i)}^{n-1}$
3. $\bigsqcup\limits_\alpha \Delta^n /{\text{face}_i{(\Delta^n_\alpha)} \sim \Delta^{n-1}_{\beta(\alpha,i)}}$ and $X$ are homeomorphic.

(Here $\bigsqcup$ is used for Disjoint union)

Some triangle-gluings which create valid top spaces can't be built with this procedure.

Also Delta-complexes are not the same as simplicial complexes, although they are quite similar.

every ∆ complex can be subdivided to be a simplicial complex. In particular, every ∆ complex is then homeo- morphic to a simplicial complex. Compared with simplicial complexes, ∆ complexes have the advantage of simpler computations since fewer simplices are required.

There are also CW complexes