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\section*{cobordism}

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

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

[[!include manifolds and cobordisms - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToCohomotopy}{Relation to Cohomotopy}\dotfill \pageref*{RelationToCohomotopy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

An (oriented) \emph{cobordism} $\Sigma$ from an (oriented) [[smooth manifold]] $X_{in}$ to an (oriented) smooth manifold $X_{out}$ is a smooth [[manifold with boundary]] $\Sigma$ such that its [[boundary]] is the [[disjoint union]]

\begin{displaymath}
\partial \Sigma \simeq X_{in} \coprod X_{out}
\end{displaymath}
with induced [[orientation]] agreeing with the given one on $X_{in}$ and being the opposite of that of $X_{out}$. Hence by labelling disjoint components of the boundary of any [[manifold with boundary]] $\Sigma$ as either ``incoming'' or ``outgoing'', then $\Sigma$ becomes a cobordism from its incoming to its outgoing boundary components.

(While $X_{in} \coprod X_{out}$ is the [[boundary]] of $\Sigma$, conversely $\Sigma$ is the ``co-boundary'' of $X_{in}\coprod X_{out}$. This is part of the reason for the ``co-'' in ``cobordism'', but sometimes one just says \emph{bordism}. The difference is more pronounced when distinguishing between [[bordism homology theory]] and [[cobordism cohomology theory]].)

With a little bit of technical conditions on the boundary inclusions added, then $(n-1)$-dimensional manifolds with [[diffeomorphism]] classes of $n$-dimensional cobordisms between them form a [[category]] (a [[category of cobordisms]]) whose [[objects]] are the manifolds, whose [[morphisms]] are the cobordisms, and whose [[composition]] operation is the operation of gluing two cobordisms along a common boundary component. Such a [[category of cobordisms]] $Bord_n$ of some [[dimension]] $n$ is naturally a [[symmetric monoidal category]] $Bord_n^{\sqcup}$ with the [[tensor product]] being the [[disjoint union]] $\sqcup$ of manifolds.

The [[connected components]] in this category are called \emph{cobordism classes} of manifolds. Under [[cartesian product]] and [[disjoint union]], these form what is called the [[cobordism ring]] in the given dimension. The study of these classes, hence of manifolds ``up to cobordisms'', is a central topic in [[algebraic topology]].

A central insight connecting [[algebraic topology]] with [[mathematical physics]] is that a [[strong monoidal functor]] on a [[category of cobordisms]] with values in something like the category [[Vect]] (with its standard [[tensor product of modules]]) may be thought of as a formalized incarnation of what in [[physics]] is called a [[topological quantum field theory]]

\begin{displaymath}
Z \colon Bord_n^\sqcup \longrightarrow Vect^\otimes
  \,.
\end{displaymath}
Here one thinks of a cobordism $\Sigma$ as a piece of [[spacetime]] (or [[worldvolume]]) of dimension $n$, and of the $(n-1)$-dimensional manifolds that this goes between as a piece of [[space]] (or [[brane]]). A functor $Z$ as above is then thought of as sending each $(n-1)$-dimensional space $X$ to its [[space of quantum states]] $Z(X)$ and each spacetime $\Sigma$ between $X_{in}$ and $X_{out}$ to a [[linear map]] $Z(\Sigma)\colon Z(X_{in}) \longrightarrow Z(X_{out})$.

From the perspective of [[mathematics]], such a functor is a way to break up [[diffeomorphism]] [[invariants]] of [[closed manifolds]] of [[dimension]] $n$ into pieces and being able to reconstruct them from gluing of data associated to manifolds with $(n-1)$-dimensional boundary.

If one considers [[manifolds with corners]] then there is a fairly evident refinement of the concept of cobordism that allows to further refine this process to gluing of [[extended cobordisms]] of any dimension $k$ going between [[extended cobordisms]] of dimension $k-1$. Such extended cobordisms of maximal dimension $n$ form a [[symmetric monoidal (infinity,n)-category]] called, naturally, the [[(infinity,n)-category of cobordisms]]. The [[cobordism hypothesis]] asserts that this is a most fundamental object in [[higher category theory]] and [[higher algebra]], namely that it is the \emph{[[free construction|free]]} [[symmetric monoidal (infinity,n)-category]] [[(infinity,n)-category with duals|with duals]]. The corresponding extended concept of [[topological quantum field theory]] is accordingly called [[extended TQFT]] or similar.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToCohomotopy}{}\subsubsection*{{Relation to Cohomotopy}}\label{RelationToCohomotopy}

The [[Pontrjagin-Thom isomorphism]] says that assigning [[Cohomotopy charge]] identifies suitable [[cobordism classes]] of [[submanifolds]] or abstract [[manifolds]] with [[cocycles]] in [[Cohomotopy|unstable Cohomotopy]] (see \href{cohomotopy#RelationToCobordismGroup}{here}) or [[stable Cohomotopy]].

The following graphics illustrates the [[Cohomotopy charge map]] of the pair creation/annihilation cobordism for 0-dimensional [[submanifolds]]:

\begin{quote}%
graphics grabbed from \href{cohomotopy+charge#SatiSchreiber19}{SS 19}


\end{quote}
(See also at \emph{[[Cohomotopy charge]] -- \href{Cohomotopy+charge#ForChargedPoints}{For charged points}}.)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[h-cobordism]]


\item [[cobordism theory]]


\item [[cobordism group]], [[cobordism ring]]


\item [[Thom spectrum]], [[Thom's theorem]]


\item [[cobordism category]]


\item [[(infinity,n)-category of cobordisms]], [[cobordism hypothesis]]


\item [[cobordism cohomology theory]]


\item [[extended cobordism]]


\item [[B-bordism]]


\item [[orbifold cobordism]]


\item [[Lagrangian cobordism]]


\item [[algebraic cobordism]]



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

\begin{itemize}%
\item [[Robert Stong]], \emph{Notes on Cobordism theory}, 1968 (\href{http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf}{toc pdf}, \href{http://press.princeton.edu/titles/6465.html}{publisher page})


\item [[Stanley Kochmann]], section 1.5 of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996


\item [[John Francis]] (notes by [[Owen Gwilliam]]), \emph{\href{http://math.northwestern.edu/~jnkf/classes/mflds/}{Topology of manifolds}}, \emph{Lecture 2: Cobordism} (\href{http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf}{pdf})


\item [[Manifold Atlas]], \emph{\href{http://www.map.mpim-bonn.mpg.de/Bordism}{Bordism}}


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Cobordism}{Cobordism}}



\end{itemize}
A relation to [[fixed point spaces]]:

\begin{itemize}%
\item Carlos Prieto, \emph{Fixed point theory and framed cobordism}, Topol. Methods Nonlinear Anal. Volume 21, Number 1 (2003), 155-169. (\href{https://projecteuclid.org/euclid.tmna/1475266278}{Euclid})

\end{itemize}
[[!redirects cobordisms]]

[[!redirects bordism]] [[!redirects bordisms]]

[[!redirects cobordant manifold]] [[!redirects cobordant manifolds]]

[[!redirects bordant manifold]] [[!redirects bordant manifolds]]

[[!redirects cobordism class]] [[!redirects cobordism classes]]



\end{document}