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We use a definition due to J. F. Adams: DEFINITION. Let G be a compact Abelian group. Let S be a closed set in N-dimensional Euclidean space and A a closed subset of S. Let m be a non-negative integer. Then there is defined the Cech homology group Hm(S, A\G)\if A is empty this is written Hm(S; G). Let K be the kernel of the inclusion homomorphism i*: Hm-i(A; G)~-»iIm_i(S; G). Let L be any subgroup of Hm-i(A ; G). Then we say that S is a surface of class §> with boundary Z)L if KZ)L. Moreover if S is a surface in the above sense but there are no closed proper subsets of S containing A which are surfaces with boundary D L then S is said to be a proper surface. It will be proved that every surface contains a proper surface. We note in passing that when A is an m — 1 sphere, dim (S) 'è-m, and G is the group of reals mod 1 this is equivalent to saying "S is a surface with boundary A iff A is not a retract of S." We take area to be the Hausdorff spherical measure f\ S. M A I N THEOREM. The minimum area of surfaces of class S with boundary Z)L is attained and if S is a proper surface of minimum area then S will be locally Euclidean at all nonboundary points at which the lower density does not exceed one, that is at almost all nonboundary points.

Solution of the Plateau problem for m-dimensional surfaces of varying topological type