topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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).
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 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
Generalization to equivariant differential topology:
Last revised on February 3, 2024 at 04:01:04. See the history of this page for a list of all contributions to it.