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\begin{document}

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\section*{differential topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{entries_in_differential_topology}{Entries in differential topology}\dotfill \pageref*{entries_in_differential_topology} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\textbf{Differential topology} is the subject devoted to the study of [[topology|topological]] properties of [[differentiable manifolds]], [[smooth manifolds]] and related [[differential geometry|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 [[smooth structure|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 [[cohomology|(co)]][[homology]] classes (e.g. [[characteristic class]]es) for differentiable manifolds and of [[differential cohomology|differential refinements of cohomology theories]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{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 manifold|symplectic structure]] and only some odd-dimensional one allow for a [[contact manifold|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 manifold|riemannian structure]] however; therefore there is no special subject of `riemannian topology'.

\hypertarget{entries_in_differential_topology}{}\subsection*{{Entries in differential topology}}\label{entries_in_differential_topology}

\begin{itemize}%
\item [[Sard's theorem]], [[transversality]], [[Thom's transversality theorem]]


\item [[Reeb sphere theorem]]


\item [[cobordism]]


\item \ldots{}



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[low dimensional topology]]


\item [[synthetic differential topology]]


\item [[equivariant differential topology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Though some of the basic results, methods and conjectures of differential topology go back to \href{http://en.wikipedia.org/wiki/Henri_Poincar%C3%A9}{Poincar\'e{}}, 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\ldots{}).

Introductions and monographs:

\begin{itemize}%
\item [[John Milnor]], \emph{Differential topology}, chapter 6 in T. L. Saaty (ed.) \emph{Lectures On Modern Mathematic II} 1964 (\href{https://archive.org/details/LecturesOnModernMathematicsIi}{web}, \href{https://ia801700.us.archive.org/6/items/LecturesOnModernMathematicsIi/Saaty-LecturesOnModernMathematicsIi.pdf}{pdf})


\item [[John Milnor]], \emph{Lectures on the h-cobordism theorem}, 1965 (\href{https://www.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf}{pdf})


\item [[James Munkres]], \emph{Elementary differential topology}, Princeton 1966


\item Andrew H. Wallace, \emph{Differential topology: first steps}, Benjamin 1968.


\item [[Victor Guillemin]], Alan Pollack, \emph{Differential topology}, Prentice-Hall 1974


\item [[Morris Hirsch]], \emph{Differential topology}, Springer GTM 33 (1976) (\href{https://link.springer.com/book/10.1007/978-1-4684-9449-5}{doi:https://link.springer.com/book/10.1007/978-1-4684-9449-5}, \href{http://books.google.com/books/about/?id=iSvnvOodWl8C}{gBooks})


\item T. Br\"o{}cker, K. J\"a{}nich, C. B. Thomas, M. J. Thomas, \emph{Introduction to differentiable topology}, 1982 (translated from German 1973 edition; $\exists$ also 1990 German 2nd edition)


\item [[Raoul Bott]], [[Loring Tu]], \emph{[[Differential Forms in Algebraic Topology]]}, Graduate Texts in Math. \textbf{82}, Springer 1982. xiv+331 pp.


\item [[John Milnor]], \emph{Topology from the differential viewpoint}, Princeton University Press, 1997. (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf}{pdf})


\item [[C. T. C. Wall]], \emph{Differential topology}, Cambridge Studies in Advanced Mathematics 154, 2016


\item [[Joel W. Robbin]], [[Dietmar Salamon]], \emph{Introduction to differential topology}, 294 pp, webdraft 2018 \href{https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf}{pdf}


\item Riccardo Benedetti, \emph{Lectures on Differential Topology} (\href{https://arxiv.org/abs/1907.10297}{arXiv:1907.10297})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Differential_topology}{Differential topology}}

\end{itemize}
Generalization to [[equivariant differential topology]]:

\begin{itemize}%
\item [[Arthur Wasserman]], \emph{Equivariant differential topology}, Topology Vol. 8, pp. 127-150, 1969 (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf}{pdf})

\end{itemize}


\end{document}