The Noether charge algebra of brane actions is typically modified by a topological anomalous term. The underlying cohomology of this anomalous term is investigated, and it is shown that the anomalous term possesses a gauge freedom. The result is that the anomalous term generates a parameterized family of topological charge algebras. When fermionic charges are taken to be nonvanishing, the known algebras underlying extended superspace formulations of the action appear in these families. This phenomena is investigated for minimal p-branes, Dp-branes and (p, q)-strings. The algebras resulting from the D-brane actions are shown to allow the construction of extended superspace actions without worldvolume gauge fields. It is shown that the actions are !-symmetric, and that the symmetry is generated by a right action. The global and local symmetry transformations of the Born-Infeld gauge field are thus shown to be described geometrically by left/right actions of the underlying extended supertranslation group. An equivalence class construction is proposed for the description of compact fermionic dimensions. In this construction, open strings in extended superspace translate to closed strings in compact superspace, and fermionic topological charges may be realized by closed strings. The differential underlying the descent construction for Noether charge algebras is shown to be naturally described as a dual of the de Rham differential. The ghost fields used in the construction are shown to be described geometrically as a vielbein with respect to this differential.