Betti  Numbers  Practice  Problems

MATH  384
Due Tuesday, April 23

1. Consider the abstract simplicial complex K defined below. What are the Betti numbers (Subscript[β, 0])(K) and (Subscript[β, 1])(K)?

k={{1},{2},{3},{4},{5},{6},{1,2},{1,4},{2,3},{2,4},{2,5},{3,5},{4,5},{4,6},{5,6},{1,2,4},{2,3,5},{4,5,6}}

2. Give an example of a simplicial complex K such that (Subscript[β, 0])(K)=3 and (Subscript[β, 1])(K)=2.

3. Give an example of a simplicial complex K such that (Subscript[β, 0])(K)=1, (Subscript[β, 1])(K)=1, and (Subscript[β, 2])(K)=1.

4. Consider  the  following set of points S. For what values ϵ>0 does the Vietoris-Rips complex (Subscript[VR, ϵ])(S) have (Subscript[β, 1])((Subscript[VR, ϵ])(S))=1?

s={{0.348214,0.0729167},{0.337202,-0.115476},{-0.451389,0.17371},{0.180853,-0.230556},{0.0873016,0.349306},{-0.508929,-0.190476},{-0.484623,0.0500992},{-0.467262,-0.255952},{-0.0957341,-0.398313},{0.116071,-0.363095},{-0.40129,0.303075},{-0.0570437,0.458829},{-0.267857,-0.38244},{0.259921,0.266865},{-0.278274,0.428571},{-0.541171,-0.0698413}}