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\begin{document}

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\section*{bordism ring}

[[!redirects cobordism ring]]

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

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

[[!include manifolds and cobordisms - contents]]

\hypertarget{cobordism_theory}{}\paragraph*{{Cobordism theory}}\label{cobordism_theory}

[[!include cobordism theory -- 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{FramedCobordism}{Framed cobordism}\dotfill \pageref*{FramedCobordism} \linebreak
\noindent\hyperlink{OrientedCobordismRing}{Oriented cobordism}\dotfill \pageref*{OrientedCobordismRing} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \textbf{(co)bordism ring} $\Omega_*=\oplus_{n\geq 0}\Omega_n$ is the [[graded ring]] whose

\begin{itemize}%
\item degree $n$ elements are classes of $n$-[[dimension|dimensional]] [[smooth manifolds]] modulo [[cobordism]];


\item product operation is given by the [[Cartesian product]] of manifolds;


\item addition operation is given by the [[disjoint union]] of manifolds.



\end{itemize}
Instead of bare manifolds one may consider manifolds with extra structure, such as [[orientation]], [[spin structure]], [[string structure]], etc. and accordingly there is

\begin{itemize}%
\item the \emph{oriented cobordism ring} $\Omega^{SO}_*$,


\item the \emph{spin cobordism ring} $\Omega^{Spin}_*$,



\end{itemize}
etc.

In this general context the bare cobordism ring is also denoted $\Omega^O_*$ or $\Omega^{un}_*$, for emphasis.

A ring [[homomorphism]] out of the cobordism ring is a (multiplicative) \emph{[[genus]]}.

More generally, for $X$ a fixed [[manifold]] there is a relative cobordism ring $\Omega_\bullet(X)$ whose

\begin{itemize}%
\item elements are classes modulo cobordism over $X$ of manifolds equipped with [[continuous functions]] to $X$ (``singular manifolds'');


\item multiplication of $[f_1 \colon \Sigma_1 \to X]$ with $[f_2 \colon \Sigma_2 \to X]$ is given by \emph{transversal intersection} $\Sigma_1 \cap_X \Sigma_2$ over $X$: perturb $f_1$ such $(f_1',f_2)$ becomes a [[transversal maps]] and then form the [[pullback]] $\Sigma_1 \times_{(f_1',f_2)} \Sigma_2$ in [[Diff]].



\end{itemize}
This product is graded in that it satisfies the \textbf{dimension formula}

\begin{displaymath}
(dim X - dim \Sigma_1) + (dim X - dim \Sigma_2) = dim X - dim (\Sigma_1 \cap_X \Sigma_2)
\end{displaymath}
hence

\begin{displaymath}
dim (\Sigma_1 \cap_X \Sigma_2 ) = (dim \Sigma_1 + dim \Sigma_2) - dim X
  \,.
\end{displaymath}
Still more generally, this may be considered for $\Sigma$ being [[manifolds with boundary]]. Then $\Omega(X,A)$ for $(X,A)$ a [[CW pair]] is the ring of cobordism classes, relative boundary, of singular manifolds $\Sigma \to X$ such that the [[boundary]] of $\Sigma$ lands in in $A$.

The resulting [[functor]]

\begin{displaymath}
(X,A) \mapsto \Omega^G_\bullet(X,A)
\end{displaymath}
constitutes a [[generalized homology theory]] (see e.g. \hyperlink{Buchstaber}{Buchstaber, II.8}). Accordingly this is called \emph{[[bordism homology theory]]}.

The [[spectrum]] that represents this under the [[Brown representability theorem]] is the universal [[Thom spectra]] $M G$ (e.g. [[MO]] for $G=O$ or [[MU]] for $G = U$), which canonically is a [[ring spectrum]] under [[Whitney sum]] of [[universal vector bundles]]. Accordingly the (co-bordism ring) itself is equivalently the bordism homology groups of the point, hence the [[stable homotopy groups]] of the [[Thom spectrum]] (this is \emph{[[Thom's theorem]]})

\begin{displaymath}
\Omega_\bullet^G \simeq \M G_\bullet(\ast) \simeq pi_\bullet(M G)
  \,.
\end{displaymath}
This remarkable relation between [[manifolds]] and [[stable homotopy theory]] is known as \emph{[[cobordism theory]]} (or ``Thom theory'').

On general grounds this is equivalently the $M G$-[[generalized cohomology]] of the point ([[cobordism cohomology theory]])

\begin{displaymath}
\Omega_\bullet^G \simeq M G^\bullet(\ast)
\end{displaymath}
which justifies calling $\Omega_\bullet^G$ both the ``bordism ring'' as well as the ``cobordism ring''.

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

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

Let $X$ be a [[smooth manifold]] of [[dimension]] $n \in \mathbb{N}$ and let $k \leq n$. Then the [[Pontryagin-Thom construction]] induces a [[bijection]]

\begin{displaymath}
[X, S^k]
  \overset{\simeq}{\longrightarrow}
  \Omega^{n-k}(X)
\end{displaymath}
from the [[cohomotopy]] sets of $X$ to the [[cobordism group]] of $(n-k)$-dimensional [[submanifolds]] with [[normal bundle|normal]] [[framed manifold|framing]] up to normally framed [[cobordism]].

In particular, the natural group structure on [[cobordism group]] (essentially given by [[disjoint union]] of submanifolds) this way induces a group structure on the cohomotopy sets.

This is made explicit for instance in \hyperlink{Kosinski93}{Kosinski 93, chapter IX}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{FramedCobordism}{}\subsubsection*{{Framed cobordism}}\label{FramedCobordism}

By [[Thom's theorem]], for any [[(B,f)-structure]] $\mathcal{B}$, there is an [[isomorphism]] (of [[commutative rings]])

\begin{displaymath}
\Omega^{\mathcal{B}}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(M\mathcal{B})
\end{displaymath}
from the [[cobordism ring]] of manifolds with stable normal $\mathcal{B}$-structure to the [[homotopy groups of a spectrum|homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]].

Now for $\mathcal{B} = Fr$ [[framing]] structure, then

\begin{displaymath}
M Fr \simeq \mathbb{S}
\end{displaymath}
is equivalently the [[sphere spectrum]]. Hence in this case [[Thom's theorem]] states that there is an isomorphism

\begin{displaymath}
\Omega^{fr}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(\mathbb{S})
\end{displaymath}
between the framed cobordism ring and the [[stable homotopy groups of spheres]].

For discussion of computation of $\pi_\bullet(\mathbb{S})$ this way, see for instance (\hyperlink{WangXu10}{Wang-Xu 10, section 2}).

For instance

\begin{itemize}%
\item $\Omega^{fr}_0 = \mathbb{Z}$ because there are two $k$-framings on a single point, corresponding to $\pi_0(O(k)) \simeq \mathbb{Z}_2$, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;


\item $\Omega^{fr}_1 = \mathbb{Z}_2$ because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to $\pi_1(O(k)) \simeq \mathbb{Z}_2$ and they are their own negatives.



\end{itemize}
\hypertarget{OrientedCobordismRing}{}\subsubsection*{{Oriented cobordism}}\label{OrientedCobordismRing}

\begin{prop}
\label{OrientedCobordismOverPoint}\hypertarget{OrientedCobordismOverPoint}{}
The cobordism ring over the point for [[orientation|oriented]] manifolds starts out as

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l}
$k$&0&1&2&3&4&5&6&7&8&$\geq 9$\\
\hline 
$\Omega^{SO}_k$&$\mathbb{Z}$&0&0&0&$\mathbb{Z}$&$\mathbb{Z}_2$&0&0&$\mathbb{Z}\oplus \mathbb{Z}$&$\neq 0$\\
\end{tabular}

\end{prop}
see e.g. (\hyperlink{ManifoldAtlas}{ManifoldAtlas})

\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[CW-complex]] (for instance a [[manifold]]), then the oriented cobordism ring is expressed in terms of the [[ordinary homology]] $H_q(X,\Omega^{SO}_{p-q})$ of $X$ with [[coefficients]] in the cobordism ring over the point, prop. \ref{OrientedCobordismOverPoint}, as

\begin{displaymath}
\Omega_p^{SO}(X)
  =
  \oplus_{q = 0}^p
  H_q(X,\Omega_{p-q}^{SO})
  \;
  mod\; odd \; torsion
  \,.
\end{displaymath}
\end{prop}
e.g. \hyperlink{ConnorFloyd62}{Connor-Floyd 62, theorem 14.2}

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

Textbook accounts include

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

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Victor Buchstaber]], \emph{Geometric cobordism theory} (\href{http://www.maths.manchester.ac.uk/~peter/MATH41101notes07.pdf}{pdf})


\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 [[Gerald Höhn]], \emph{Komplexe elliptische Geschlechter und $S^1$-\"a{}quivariante Kobordismustheorie} (german) (\href{http://arxiv.org/PS_cache/math/pdf/0405/0405232v1.pdf}{pdf})


\item [[Sander Kupers]], \emph{Oriented bordism: Calculation and application} (\href{http://web.stanford.edu/~kupers/orientedbordism.pdf}{pdf})



\end{itemize}
Details for framed cobordism are spelled out in

\begin{itemize}%
\item [[Guozhen Wang]], Zhouli Xu, section 2 of \emph{A survey of computations of homotopy groups of Spheres and Cobordisms}, 2010 (\href{http://math.mit.edu/~guozhen/homotopy%20groups.pdf}{pdf})


\item [[Andrew Putman]], \emph{Homotopy groups of spheres and low-dimensional topology} (\href{http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf}{pdf})



\end{itemize}
The relation to [[cohomotopy]] is made explicit in

\begin{itemize}%
\item [[Antoni Kosinski]], chapter IX of \emph{Differential manifolds}, Academic Press 1993 (\href{http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf}{pdf})

\end{itemize}
Further discussion of oriented cobordism includes

\begin{itemize}%
\item Manifold Atlas, \emph{\href{http://www.map.mpim-bonn.mpg.de/Oriented_bordism}{Oriented bordism}}


\item P. E. Conner, E. E. Floyd, \emph{Differentiable periodic maps}, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86. (\href{http://projecteuclid.org/euclid.bams/1183524501}{Euclid}, \href{http://www.maths.ed.ac.uk/~aar/papers/cf1.pdf}{pdf})



\end{itemize}
A historical review in the context of [[complex cobordism cohomology theory]]/[[Brown-Peterson theory]] is in

\begin{itemize}%
\item [[Doug Ravenel]], chapter 4, section 2 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]}

\end{itemize}
On [[fibered cobordism groups]]:

\begin{itemize}%
\item Astey, Greenberg, Micha, Pastor, \emph{Some fibered cobordisms groups are not finitely generated} (\href{http://www.ams.org/journals/proc/1988-104-01/S0002-9939-1988-0958093-3/S0002-9939-1988-0958093-3.pdf}{pdf})

\end{itemize}
Discussion of the $G$-equivariant complex coborism ring includes

\begin{itemize}%
\item G. Comezana and [[Peter May]], \emph{A completion theorem in complex cobordism}, in Equivariant Homotopy and Cohomology Theory, CBMS Regional conference series in Mathematics, American Mathematical Society Publications, Volume 91, Providence, 1996.


\item [[Igor Kriz]], \emph{The $\mathbb{Z}/p$--equivariant complex cobordism ring}, from: ``Homotopy invariant algebraic structures (Baltimore, MD, 1998)'', Amer. Math. Soc. Providence, RI (1999) 217--223


\item [[Neil Strickland]], \emph{Complex cobordism of involutions}, Geom. Topol. 5 (2001) 335-345 (\href{http://arxiv.org/abs/math/0105020}{arXiv:math/0105020})


\item William Abrams, \emph{Equivariant complex cobordism}, 2013 (\href{http://deepblue.lib.umich.edu/bitstream/handle/2027.42/99796/abramwc_1.pdf}{pdf})



\end{itemize}
[[!redirects bordism ring]]

[[!redirects cobordism rings]]

[[!redirects cobordism group]] [[!redirects cobordism groups]]



\end{document}