e2 e1 e5 e4 e3 v1 v2 v3 v4 f The dimension of f = 2 The dimension of ei = 1 The dimension of vi = 0 The boundary of vi = 0 The boundary of f = e1 + e1 + e3 The boundary of e1 = v2 – v1 The boundary of e2 = v3 – v2 The boundary of e4 = v1 – v4 The boundary of e5 = v4 – v3 The boundary of e1 + e2 = v3 – v1 The boundary of e1 + e4 = v2 – v4 The boundary of e1 + e5 = v2 – v1 + v4 – v3 The boundary of e1 + e2 + e3 = 0 The boundary of e3 + e4 + e5 = 2v1 - 2v3 The boundary of -e3 + e4 + e5 = 0 The boundary of e1 + e2 + e4 + e5 = 0 Let X = e1 + e2 + e3, let Y = -e3 + e4 + e5, and let Z = e1 + e2 + e4 + e5. Show that Z = X + Y X + Y = (e1 + e2 + e3) + (-e3 + e4 + e5) = e1 + e2 + e3 – e3 + e4 + e5 = e1 + e2 + e4 + e5 = Z 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: e1 + e2 + e3 e3 + e4 + e5 e1 + e2 + e4 + e5 Note: any sum of cycles is a cycle. List four 0-dimensional cycles: v1, v2, v3, v4 Are there any 2-dimensional cycles? No There is only 1 face and its boundary is not 0 The boundary of f is the cycle = e1 + e2 + e3 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? 4 How many independent 1-dimensional cycles does this complex have? 3 How many independent 2-dimensional cycles does this complex have? 1 f v4 v3 v1 v2