higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Differential topology is the subject devoted to the study of topological properties of smooth manifolds and closely related spaces like stratifolds, orbifolds and more general differentiable stacks. It is also concerned with the problem of finding out which topological (or PL) manifolds allow a differentiable structure and the degree of nonuniqueness of that structure if they do. It is also concerned with concrete constructions of (co)homology classes (e.g. characteristic classes) for differentiable manifolds and of differential refinements of cohomology theories.
Many considerations, and classification problems, depend crucially on dimension, and the case of high-dimensional manifolds (the notion of ‘high’ depends on the problem) is often very different from the situation in each of the low dimensions; thus there are specialists’ subjects like $3$-(dimensional) topology and $4$-topology. There are restrictions on an underlying topology which is allowed for some sorts of additional structures on a differentiable manifold.
For example, only some even-dimensional differentiable manifolds allow for symplectic structure and only some odd-dimensional one allow for a contact structure; in these cases moreover special constructions of topological invariants like Floer homology and symplectic field theory exist.
This yields the relatively young subjects of symplectic and contact topologies, with the first significant results coming from Gromov. Any (Hausdorff paracompact finite-dimensional) differentiable manifold allows for riemannian structure however; therefore there is no special subject of ‘riemannian topology’.
Though some of the basic results, methods and conjectures of differential topology go back to Poincaré, Whitney, Morse and Pontrjagin, it became an independent field only in the late 1950s and early 1960s with the seminal works of Smale, Thom, Milnor and Hirsch. Soon after the initial effort on foundations, mainly in American school, a strong activity started in Soviet Union (Albert Schwarz, A. S. Mishchenko, S. Novikov, V. A. Rokhlin, M. Gromov…). * Morris W. Hirsch, Differential topology, Springer GTM 33, gBooks * T. Bröcker, K. Jänich, C. B. Thomas, M. J. Thomas, Introduction to differentiable topology, 1982 (translated from German 1973 edition; $\exists$ also 1990 German 2nd edition) * R. Bott, L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Math. 82, Springer 1982. xiv+331 pp.