Differential topology vs differential geometry
nLabdifferential topologyContents
IdeaDifferential topology is the subject devoted to the study of topological properties of differentiable manifolds, smooth manifolds and related differential geometric spaces such as stratifolds, orbifolds and more generally differentiable stacks. Differential topology 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 (e.g. exotic smooth structures). 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. ExamplesMany 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 33-(dimensional) topology and 44-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. Entries in differential topology
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ReferencesThough 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 the American school, a strong activity started in Soviet Union (Albert Schwarz, A. S. Mishchenko, S. Novikov, V. A. Rokhlin, M. Gromov). Introductions and monographs:
Survey with connections to algebraic topology:
See also
Generalization to equivariant differential topology:
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