§ Simplicial approxmation of maps (TODO)
§ What we want:
every map is homotopic to a simplicial map : ie, a map that sends vertices to vertices, and sends other points through an extension of linearity. such that where is the simplicial approximation of .
Recall that is the intersection of interiors of all simplices that
contain the vertex . So on a graph, it's going to be a "star shaped" region
around the vertex of all the edges around the vertex.
§ Why this can't happpen
Consider a triangle as a simplex of a circle. We want to represent rotations of
the circle. I can rotate around a circle once, twice, thrice, ... As many
times as I want. However, if all I have is a triangle, I can represent rotating
once as the map and rotating twice as
maybe , but that's it. I've run out of
room! So I need to subdivide the simplex to get "more points" to represent
§ The correct statement