e2 e1 e5 e4 e3 v1 v2 v3 v4 f The dimension of f = The dimension of ei = The dimension of vi = The boundary of vi = The boundary of f = The boundary of e1 = The boundary of e2 = The boundary of e4 = The boundary of e5 = The boundary of e1 + e2 = The boundary of e1 + e4 = The boundary of e1 + e5 = The boundary of e1 + e2 + e3 = The boundary of e3 + e4 + e5 = The boundary of -e3 + e4 + e5 = The boundary of e1 + e2 + e4 + e5 = Let X = e1 + e2 + e3, let Y = -e3 + e4 + e5, and let Z = e1 + e2 + e4 + e5. Show that Z = X + Y f

e2 e1 e5 e4 e3 v1 v2 v3 v4 Note z is a cycle if the boundary of z = 0. List three 1-dimensional cycles: List four 0-dimensional cycles: Are there any 2-dimensional cycles Note the simplicial complex on the bottom right is the boundary of a tetrahedron (and thus topologically equivalent to a sphere). How many independent 0-dimensional cycles does this complex have? How many independent 1-dimensional cycles does this complex have? How many independent 2-dimensional cycles does this complex have? f v4 v3 v1 v2