Almgren's isomorphism theorem

Almgren isomorphism theorem is a result in geometric measure theory and algebraic topology about the topology of the space of flat cycles in a Riemannian manifold.

The theorem plays a fundamental role in the Almgren–Pitts min-max theory as it establishes existence of topologically non-trivial families of cycles, which were used by Frederick J. Almgren Jr., Jon T. Pitts and others to prove existence of (possibly singular) minimal submanifolds in every Riemannian manifold. In the special case of codimension 1 cycles with mod 2 coefficients Almgren isomorphism theorem implies that the space of cycles is weakly homotopy equivalent to the infinite real projective space.