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\section*{algebraic cobordism}

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_2nngraded_part}{The (2n,n)-graded part}\dotfill \pageref*{the_2nngraded_part} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{the_oriented_cohomology_theory}{The oriented cohomology theory}\dotfill \pageref*{the_oriented_cohomology_theory} \linebreak


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

\emph{Algebraic cobordism} is the bigraded [[generalized cohomology theory]] represented by the [[motivic Thom spectrum]] $MGL$. Hence it is the algebraic or [[motives|motivic]] analogue of [[complex cobordism]]. The $(2n,n)$-graded part has a geometric description via [[cobordism]] classes, at least over fields of characteristic zero.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $S$ be a [[scheme]] and $MGL_S$ the [[motivic Thom spectrum]] over $S$. \textbf{Algebraic cobordism} is the [[generalized motivic cohomology theory]] $MGL_S^{*,*}$ represented by $MGL_S$:

\ldots{} formula here \ldots{}

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

\hypertarget{the_2nngraded_part}{}\subsubsection*{{The (2n,n)-graded part}}\label{the_2nngraded_part}

Let $S = Spec(k)$ where $k$ is a field of characteristic zero. A geometric description of the $(2n,n)$-graded part of algebraic cobordism was given by [[Marc Levine]] and [[Fabien Morel]]. More precisely, \hyperlink{LevineMorel}{Levine-Morel} constructed the universal \emph{oriented cohomology theory} $\Omega^* : \Sm_k \to CRing^*$. Here \emph{oriented} signifies the existence of [[direct image]] or [[Gysin homomorphisms]] for [[proper morphisms of schemes]]. This implies the existence of[[Chern classes]] for [[vector bundles]].

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(\hyperlink{LevineMorel}{Levine-Morel})}. There is a canonical isomorphism of graded rings

\begin{displaymath}
\mathbf{L}^* \stackrel{\sim}{\longrightarrow} \Omega^*(\Spec(k))
\end{displaymath}
where $\mathbf{L}^*$ denotes the [[Lazard ring]] with an appropriate grading.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(\hyperlink{LevineMorel}{Levine-Morel})}. Let $i : Z \hookrightarrow X$ be a [[closed immersion]] of smooth $k$-schemes and $j : U \hookrightarrow X$ the complementary [[open immersion]]. There is a canonical exact sequence of graded abelian groups

\begin{displaymath}
\Omega^{*-d}(Z) \stackrel{i_*}{\to} \Omega^*(X) \stackrel{j^*}{\to} \Omega^*(U) \to 0,
\end{displaymath}
where $d = \codim(Z, X)$.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(\hyperlink{LevineMorel}{Levine-Morel})}. Given an embedding $k \hookrightarrow \mathbf{C}$, the canonical homomorphism of graded rings

\begin{displaymath}
\Omega^*(k) \longrightarrow MU^{2*}(pt)
\end{displaymath}
is invertible.

\end{theorem}
\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(\hyperlink{Levine2008}{Levine 2008})}. The canonical homomorphisms of graded rings

\begin{displaymath}
\Omega^*(X) \longrightarrow MGL^{2*,*}(X)
\end{displaymath}
are invertible for all $X \in \Sm_k$.

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

\begin{itemize}%
\item [[motivic Thom spectrum]]
\item [[complex cobordism]]
\item [[motivic homotopy theory]]
\item [[motivic cohomology]]
\item [[algebraic K-theory]]

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

\begin{itemize}%
\item [[Vladimir Voevodsky]], \emph{$\mathbf{A}^1$-Homotopy Theory}, Doc. Math., Extra Vol. ICM 1998(I), 417-442, \href{http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/00/Voevodsky.MAN.html}{web}.


\item [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], \emph{A universality theorem for Voevodsky's algebraic cobordism spectrum}, Homology, Homotopy and Applications, 2008, 10(2), 211-226, \href{http://arxiv.org/abs/0709.4116}{arXiv}.


\item [[Ivan Panin]], K. Pimenov, [[Oliver Röndigs]], \emph{On the relation of Voevodsky's algebraic cobordism to Quillen's K-theory}, Inventiones mathematicae, 2009, 175(2), 435-451, \href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.7301}{DOI}, \href{http://arxiv.org/abs/0709.4124}{arXiv}.


\item [[Markus Spitzweck]], \emph{Algebraic cobordism in mixed characteristic}, \href{http://arxiv.org/abs/1404.2542}{arXiv}.


\item [[Marc Hoyois]], \emph{From algebraic cobordism to motivic cohomology}, \href{http://math.mit.edu/~hoyois/papers/hopkinsmorel.pdf}{pdf}, \href{http://arxiv.org/abs/1210.7182}{arXiv}.


\item [[Marc Levine]], [[Girja Shanker Tripathi]], \emph{Quotients of MGL, their slices and their geometric parts}, \href{http://arxiv.org/abs/1501.02436}{arXiv:1501.02436}.



\end{itemize}
A discussion on MathOverflow:

\begin{itemize}%
\item \emph{Interdependence between A{\tt \symbol{94}}1-homotopy theory and algebraic cobordism}, \href{http://mathoverflow.net/questions/36659/interdependence-between-a1-homotopy-theory-and-algebraic-cobordism/36698#36698}{MO/36659}.

\end{itemize}
\hypertarget{the_oriented_cohomology_theory}{}\subsubsection*{{The oriented cohomology theory}}\label{the_oriented_cohomology_theory}

\begin{itemize}%
\item [[Marc Levine]], [[Fabien Morel]], \emph{Cobordisme alg\'e{}brique I}, Note aux C.R. Acad. Sci. Paris, 332 S\'e{}rie I, p. 723--728, 2001 (\href{http://dx.doi.org/10.1016/S0764-4442(01}{doi}01833-X)); \emph{Cobordisme alg\'e{}brique II}, Note aux C.R. Acad. Sci. Paris, 332 S\'e{}rie I, p. 815--820, 2001 (\href{http://dx.doi.org/10.1016/S0764-4442(01}{doi}01832-8)).


\item [[Marc Levine]], [[Fabien Morel]], \emph{Algebraic cobordism}, Springer 2007, \href{https://www.uni-due.de/~bm0032/publ/AlgCobordBook4.pdf}{pdf}.


\item [[Marc Levine]], \emph{A survey of algebraic cobordism}, \href{http://www.uni-due.de/~bm0032/publ/SurveyAlgCobord.pdf&#8206;}{pdf}


\item [[Marc Levine]], \emph{Algebraic cobordism}, Proceedings of the ICM, Beijing 2002, vol. 2, 57--66, \href{http://arxiv.org/abs/math/0304206}{math.KT/0304206}


\item [[Marc Levine]], \emph{Three lectures on algebraic cobordism}, University of Western Ontario Mathematics Department, 2005, \href{http://www.uni-due.de/~bm0032/publ/CobordismLec1LS.pdf}{Lecture I}, \href{http://www.uni-due.de/~bm0032/publ/CobordismLec2LS.pdf}{Lecture II}, \href{http://www.uni-due.de/~bm0032/publ/CobordismLec4LS.pdf}{Lecture III}.


\item M. Levine, F. Morel, [[Oberwolfach]] Arbeitsgemeinschaft mit aktuellem Thema, April 2005 \href{http://www.ems-ph.org/journals/show_abstract.php?issn=1660-8933&vol=2&iss=2&rank=2}{report}, \href{http://www.ems-ph.org/journals/show_pdf.php?issn=1660-8933&vol=2&iss=2&rank=2}{notes}



\end{itemize}
A simpler construction was given in

\begin{itemize}%
\item M. Levine, R. Pandharipande, \emph{Algebraic cobordism revisited}, \href{http://arxiv.org/abs/math/0605196}{math.AG/0605196}

\end{itemize}
A [[Borel-Moore homology]] version of $MGL^{*,*}$ is considered in

\begin{itemize}%
\item [[Marc Levine]], \emph{Oriented cohomology, Borel-Moore homology and algebraic cobordism}, \href{http://arxiv.org/abs/0807.2257}{arXiv}.

\end{itemize}
The comparison with $MGL^{2*,*}$ is in

\begin{itemize}%
\item [[Marc Levine]], \emph{Comparison of cobordism theories}, Journal of Algebra, 322(9), 3291-3317, 2009, \href{http://arxiv.org/abs/0807.2238}{arXiv}.

\end{itemize}
The construction was extended to [[derived algebraic geometry|derived schemes]] in the paper

\begin{itemize}%
\item [[Parker Lowrey]], [[Timo Schuerg]]. \emph{Derived algebraic bordism}, 2012, \href{http://arxiv.org/abs/1211.7023}{arXiv:1211.7023}.

\end{itemize}
The close connection of algebraic cobordism with [[K-theory]] is discussed in

\begin{itemize}%
\item Jos\'e{} Luis Gonz\'a{}lez, Kalle Karu. \emph{Universality of K-theory}. 2013. \href{http://arxiv.org/abs/1301.3815}{arXiv:1301.3815}.

\end{itemize}
An algebraic analogue of h-cobordism:

\begin{itemize}%
\item Aravind Asok, Fabien Morel, \emph{Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms}, \href{http://arxiv.org/abs/0810.0324}{arXiv:0810.0324}

\end{itemize}


\end{document}