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\begin{document}

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\section*{cobordism theory}

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

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The concept of \emph{[[cobordism]]} sits at a subtle connection between [[differential topology]]/[[differential geometry]] and [[stable homotopy theory]]/[[higher category theory]], this is the content of what is often called ``cobordism theory''.

The insight goes back to the seminal thesis (\hyperlink{Thom54}{Thom 54}), which established that the [[Pontryagin-Thom construction]] exhibits the [[cobordism ring]], whose elements are cobordism [[equivalence classes]] of [[manifolds]], as the [[homotopy groups]] of a [[spectrum]], whence called the \emph{[[Thom spectrum]]}.

Via the [[Brown representability theorem]], the full [[Thom spectrum]] represents a [[generalized cohomology theory]], whence called \emph{[[cobordism cohomology theory]]}. For every kind of topological structure carried by manifolds this has a variant, such as notably [[complex cobordism cohomology theory]]. These Thom spectra and their cobordism cohomology theories play a special role in [[stable homotopy theory]], for instance for the concepts of \emph{[[orientation in generalized cohomology]]} and the concept of \emph{[[genus]]}.

A more recent refinement of this statement is (the proof of) the [[cobordism hypothesis]] which identified the ([[framed manifold|framed]]) [[(∞,n)-category of cobordisms]] as the free [[symmetric monoidal (∞,n)-category with duals]] on a single object.

For more introduction see at \emph{[[Introduction to Cobordism and Complex Oriented Cohomology]]}.

[[!include cobordism theory -- contents]]

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

Original articles include

\begin{itemize}%
\item [[René Thom]], \emph{Quelques propri\'e{}t\'e{}s globales des vari\'e{}t\'e{}s diff\'e{}rentiables} Comment. Math. Helv. 28, (1954). 17-86


\item [[Michael Atiyah]], \emph{Thom complexes}, Proc. London Math. Soc. (3), 11:291--310, 1961. 10



\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Robert Stong]], \emph{Notes on cobordism theory} , Princeton Univ. Press (1968) (\href{http://www.maths.ed.ac.uk/~aar/papers/stongcob.pdf}{pdf})


\item [[Stanley Kochman]], chapters I and IV of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[Yuli Rudyak]], \emph{On Thom spectra, orientability and cobordism}, Springer Monographs in Mathematics, 1998 (\href{http://www.maths.ed.ac.uk/~aar/papers/rudyakthom.pdf}{pdf})



\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Tom Weston]], \emph{An introduction to cobordism theory} (\href{http://people.math.umass.edu/~weston/oldpapers/cobord.pdf}{pdf})


\item [[John Francis]], \emph{Topology of manifolds} course notes (2010) (\href{http://math.northwestern.edu/~jnkf/classes/mflds/}{web}), Lecture 2 \emph{Cobordisms} (notes by [[Owen Gwilliam]]) (\href{http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf}{pdf}), Lecture 3 \emph{Thom's theorem} (notes by A. Smith) (\href{http://math.northwestern.edu/~jnkf/classes/mflds/3thom.pdf}{pdf})


\item [[Cary Malkiewich]], \emph{Unoriented cobordism and $M O$}, 2011 (\href{http://math.uiuc.edu/~cmalkiew/cobordism.pdf}{pdf})


\item [[Johannes Ebert]], \emph{A lecture course on Cobordism Theory}, 2012 (\href{http://wwwmath.uni-muenster.de/u/jeber_02/skripten/bordism-skript.pdf}{pdf})


\item \emph{[[Introduction to Cobordism and Complex Oriented Cohomology]]}



\end{itemize}
This one here includes the connection to the [[(infinity,n)-category of cobordisms]]

\begin{itemize}%
\item [[Dan Freed]], \emph{Bordism: old and new}, 2013 (\href{http://www.ma.utexas.edu/users/dafr/bordism.pdf}{pdf})

\end{itemize}
For [[complex cobordism cohomology]] see there and see

\begin{itemize}%
\item [[Frank Adams]], part I, part II of \emph{[[Stable homotopy and generalized homology]]}, Chicago Lectures in mathematics, 1974

\end{itemize}


\end{document}