# Differential topology

Important general mathematical concepts were developed in differential topology. These include fibrations and bundles over a space, and the differential-geometric and topological concepts connected with it: connections, $G$-structures, characteristic classes, and also rigged (framed) manifolds. Of major importance in the development of differential topology was the theory of (co)bordisms, with its several applications in algebraic and analytical geometry (the Riemann–Roch theorem), the theory of elliptic operators (the index theorem), and also in topology itself. The 1950s witnessed the discovery of various different smooth structures on spheres; this was followed by the classification of manifolds having the homotopy type of the spheres and by the proof of the generalized Poincaré conjecture: The solution was found for the problem of finding a complete system of diffeomorphism invariants of all simply-connected manifolds (of a dimension not less than 5). In the 1960s fundamental topological problems were solved by applying the methods of differential topology: It was found that the characteristic classes of real manifolds were topologically invariant; the relation between the categories of differentiable, piecewise-linear and topological manifolds were clarified; the methods of classification of smooth manifolds were generalized to include non-simply-connected manifolds (though admittedly not very effectively); and algebraic $K$-theory and Hermitian $K$-theory were created. For non-simply-connected manifolds, fundamental relationships were discovered between the characteristic classes and Hermitian forms over the fundamental group of the manifold and the homology. Subsequently, fundamental results were obtained by methods of functional analysis and by algebraic methods, concerning the homotopy invariance of classes and the theory of Hermitian forms over cochains with an involution. Of major importance are methods concerning the problem of classification of imbeddings of one manifold into another and its various generalizations.
A separate branch of differential topology, related to the calculus of variations, is the global theory of extremals of various functionals on manifolds of geodesics. It strongly influenced the development of topology itself by making possible a classification of vector bundles and, subsequently, by producing a method of studying the topological invariants, provided by $K$-theory. Multi-dimensional global problems of the calculus of variations on manifolds proved to be more difficult; the problem which was principally studied was the problem of minimal surfaces, as the extremals of Dirichlet-type functionals. In the 1970s the theory of elementary particles gave rise to several essentially new problems in multi-dimensional variational calculus.