# definition of a $\Delta$ - complex

I've been given this image from hatchers algebraic topology as an example of a $\Delta$ complex but with the explicit definition as follows

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_\alpha$ : $\Delta{^n}→X$ , with n depending on the index $\alpha$, such that:

(i) The restriction $\sigma_\alpha$ | $\mathring{\Delta^{n}}$ is injective, and each point of X is in the image of exactly one such restriction $\sigma_\alpha$ | $\mathring{\Delta^{n}}$

(ii) Each restriction of $\sigma_\alpha$ to a face of $\Delta^n$ is one of the maps $\sigma_\beta$ :$\Delta^{n-1}$→X . Here we are identifying the face of $\Delta^n$ with $\Delta^{n-1}$ by the canonical linear homeomorphism between them that preserves the ordering of the vertices.

(iii) A set A⊂X is open iff $\sigma^{-1}_\alpha$(A) is open in $\Delta^{n}$ $\forall \sigma_\alpha$

what i don't understand is how i would get the explicit $\Delta$-Complex from this diagram, does anyone have any recommended resources that would have some kind of step by step example showing how to find the $\Delta$ complex of a torus.

note: i plan to use the torus as an elementary example to find other $\Delta$-Complexes for standard shapes ## 1 Answer

There's one 0-simplex, three 1-simplices, and 2 2-simplices in this diagram. $U$ and $L$ are the images of the 2-simplex, $a,b,c$ are the images of the 1-simplex, and $v$ is the image of the zero-simplex. Is there something else causing you trouble here?

• ah i see so the $\Delta$ complex is made up of the 0, 1 and 2 simplifies, thank you! – Peter A Jan 14 '15 at 0:30
• Yes, technically, of the maps from the $0,1,2$-simplices. Glad to have helped. – Kevin Arlin Jan 14 '15 at 0:32
• @KevinCarlson can you see my question here it is slightly related math.stackexchange.com/questions/1701168/… – user111750 Mar 17 '16 at 3:59