The h-chains admit a boundary operator Δ Δ C x , H ( h ) defined as the (h-1)-simplex forming the boundary of the h-chain. A coboundary operator ∇ ∇ C x , H ( h ) can be similarly defined to yield a (h+1)-chain. The dual space of chains consists of h -cochains Φ Φ ( h ) ( C ( h ) ) , which are linear functions acting on the h-chains mapping them to real numbers. The boundary and coboundary operators admit similar structures in dual space called the dual boundary Δ Δ ^ ^ and dual coboundary ∇ ∇ ^ ^ defined to satisfy