nLab differential topology

Idea

Differential topology is the subject devoted to the study of algebro-topological and homotopy-theoretic properties of differentiable manifolds, smooth manifolds and related differential geometric spaces such as stratifolds, orbifolds and more generally differentiable stacks.

A key part of differential topology is cobordism theory, where the Pontryagin-Thom theorem relates the stable homotopy theory of Thom spectra to cobordism classes of smooth (sub-)manifolds (for instance cohomotopy to normally framed cobordism).

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.

More recently, the smooth Oka principle reveals a deep structure in differential topology which is visible in the full generality of higher differential geometry (smooth $\infty$-stacks).

Examples

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’.

Entries in differential topology

• Hopf degree theorem, equivariant Hopf degree theorem

• Poincaré–Hopf theorem

• Pontrjagin-Thom theorem

• Sard's theorem, transversality, Thom's transversality theorem

• Reeb sphere theorem

• cobordism

• …

Related entries

• low dimensional topology

• synthetic differential topology

• equivariant differential topology

• surgery

References

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 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:

• John Milnor, Differential topology, chapter 6 in T. L. Saaty (ed.) Lectures On Modern Mathematic II 1964 (pdf)

• John Milnor, Lectures on the h-cobordism theorem, 1965 (pdf)

• James R. Munkres, Elementary Differential Topology, Annals of Mathematics Studies 54 (1966), Princeton University Press (doi:10.1515/9781400882656).

• Andrew H. Wallace, Differential topology: first steps, Benjamin 1968.

• Victor Guillemin, Alan Pollack, Differential topology, Prentice-Hall 1974

• Morris Hirsch, Differential topology, Springer Graduate Texts in Mathematics 33 (1976) [doi:10.1007/978-1-4684-9449-5, gBooks]

• Theodor Bröcker, Klaus Jänich, Introduction to differentiable topology (1982) [ISBN:9780521284707]

  (translated from the German 1973 edition)

• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology, Graduate Texts in Math. 82, Springer 1982. xiv+331 pp.

• John Milnor, Topology from the differential viewpoint, Princeton University Press, 1997. (ISBN:9780691048338, pdf)

• Mladen Bestvina (notes by Adam Keenan), Differentiable Topology and Geometry, 2002 (pdf, pdf)

• C. T. C. Wall, Differential topology, Cambridge Studies in Advanced Mathematics 154, 2016

• Joel W. Robbin, Dietmar Salamon, Introduction to differential topology, 294 pp, webdraft 2018 pdf

• Riccardo Benedetti, Lectures on Differential Topology, Graduate Studies in Mathematics 218, AMS 2021 (arXiv:1907.10297, ISBN: 978-1-4704-6674-9)

Survey with connections to algebraic topology:

• Sergei Novikov, Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (doi:10.1007/978-3-662-10579-5, pdf)

• Jean Dieudonné, A History of Algebraic and Differential Topology, 1900 - 1960, Modern Birkhäuser Classics 2009 (ISBN:978-0-8176-4907-4)

See also

• Wikipedia, Differential topology

Generalization to equivariant differential topology:

• Arthur Wasserman, Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)