1. 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

2. 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