mathematical physics - Does Clifford algebra depend on the topology of manifold?

We know the greatest feature of Clifford algebra is coordinate-free. One can do vector operations without knowing the representation of vectors. And due to its very characteristc, Clifford or geometric algebra is believed to a reinterpretation of differential geometry suggested mainly by Hestenes and Doran. But as far as I know, many manifold-related theorem depends on the topology of the manifold such as connectedness, compactness, boundaryless or not. I want to know how Clifford algebra behave in different topologies?