cobordism theory = manifolds and cobordisms + stable homotopy theory/higher category theory
Concepts of cobordism theory
Pontrjagin's theorem (equivariant, twisted):
$\phantom{\leftrightarrow}$ Cohomotopy
$\leftrightarrow$ cobordism classes of normally framed submanifolds
$\phantom{\leftrightarrow}$ homotopy classes of maps to Thom space MO
$\leftrightarrow$ cobordism classes of normally oriented submanifolds
complex cobordism cohomology theory
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
Algebraic cobordism is the bigraded generalized cohomology theory represented by the motivic Thom spectrum $MGL$. Hence it is the algebraic or motivic analogue of complex cobordism. The $(2n,n)$-graded part has a geometric description via cobordism classes, at least over fields of characteristic zero.
Let $S$ be a scheme and $MGL_S$ the motivic Thom spectrum over $S$. Algebraic cobordism is the generalized motivic cohomology theory? $MGL_S^{*,*}$ represented by $MGL_S$:
… formula here …
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, Levine-Morel constructed the universal oriented cohomology theory $\Omega^* : \Sm_k \to CRing^*$. Here oriented signifies the existence of direct image or Gysin homomorphisms for proper morphisms of schemes. This implies the existence ofChern classes for vector bundles.
(Levine-Morel). There is a canonical isomorphism of graded rings
where $\mathbf{L}^*$ denotes the Lazard ring with an appropriate grading.
(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
where $d = \codim(Z, X)$.
(Levine-Morel). Given an embedding $k \hookrightarrow \mathbf{C}$, the canonical homomorphism of graded rings
is invertible.
(Levine 2008). The canonical homomorphisms of graded rings
are invertible for all $X \in \Sm_k$.
flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:
bordism theory$\;$M(B,f) (B-bordism):
relative bordism theories:
global equivariant bordism theory:
algebraic:
There are two notions of “algebraic cobordism”, not closely related, one due to Snaith 77, and one due to Levine-Morel 01.
Victor Snaith, Towards algebraic cobordism, Bull. Amer. Math. Soc. 83 (1977), 384-385 (doi:10.1090/S0002-9904-1977-14281-X)
Victor Snaith, Algebraic Cobordism and K-theory, Mem. Amer. Math. Soc. no 221 (1979)
David Gepner, Victor Snaith, On the motivic spectra representing algebraic cobordism and algebraic K-theory, Documenta Math. 2008 (arXiv:0712.2817)
The construction in Snaith 77, motivated from the Conner-Floyd isomorphism, uses a variant of his general construction (Snaith's theorem) of a periodic multiplicative cohomology theory $X(b)^*(-)$ out of a pair consisting of a homotopy commutative H-monoid? $X$ and a class $b\in \pi_n(X)$:
When $X = B S^1$ (the classifying space of the circle group) and $b$ is a generator of $\pi_2(BS^1)\cong\mathbb{Z}$ then $X(b)^*(-)$ is isomorphic with 2-periodic complex K-theory.
When $X = B U$ and $b$ a generator of $\pi_2(BU)\cong\mathbb{Z}$ one obtains $MU^*[u_2,u_2^{-1}]$ where MU is the (topological) complex cobordism cohomology and $u_2$ is the periodicity element.
Then Snaith introduces a variant of such constructions with a more general ring $A$ replacing the complex numbers; and uses the Quillen’s description of algebraic K-theory of a ring $A$ in terms of the classifying space $B GL(A)$; this way he obtains an algebraic cobordism theory.
Later, Gepner-Snaith 08 returned to the question of algebraic cobordism this time using the motivic version of algebraic cobordism of Voevodsky, namely the motivic spectrum $M GL$ representing universal oriented motivic cohomology theory (which is different from Morel-Voevodsky algebraic cobordism), and to the motivic version of Conner-Floyd isomorphism for which they give a comparably short proof.
Marc Levine, Fabien Morel, Cobordisme algébrique I, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 723–728, 2001 (doi:10.1016/S0764-4442(01)01833-X); Cobordisme algébrique II, Note aux C.R. Acad. Sci. Paris, 332 Série I, p. 815–820, 2001 (doi:10.1016/S0764-4442(01)01832-8).
Marc Levine, Algebraic cobordism, Proceedings of the ICM, Beijing 2002, vol. 2, 57–66, math.KT/0304206
Marc Levine, Fabien Morel, Algebraic cobordism, Springer 2007, pdf.
Marc Levine, A survey of algebraic cobordism (pdf)
Marc Levine, Three lectures on algebraic cobordism, University of Western Ontario Mathematics Department, 2005, Lecture I, Lecture II, Lecture III.
Marc Levine, Fabien Morel, Oberwolfach Arbeitsgemeinschaft mit aktuellem Thema, April 2005 report, notes
Ivan Panin, K. Pimenov, Oliver Röndigs, A universality theorem for Voevodsky’s algebraic cobordism spectrum, Homology, Homotopy and Applications, 2008, 10(2), 211-226 (arXiv:0709.4116)
Ivan Panin, K. Pimenov, Oliver Röndigs, On the relation of Voevodsky’s algebraic cobordism to Quillen’s K-theory, Inventiones mathematicae, 2009, 175(2), 435-451, doi:10.1.1.244.7301, arXiv:0709.4124.
Marc Hoyois, From algebraic cobordism to motivic cohomology, pdf, arXiv.
Markus Spitzweck, Algebraic cobordism in mixed characteristic (arXiv:1404.2542)
Marc Levine, Girja Shanker Tripathi?, Quotients of MGL, their slices and their geometric parts, arXiv:1501.02436.
More chat about the relation to motivic homotopy theory:
A simpler construction was given in
A Borel-Moore homology version of $MGL^{*,*}$ is considered in
The comparison with $MGL^{2*,*}$ is in
The construction was extended to derived schemes in the paper
The close connection of algebraic cobordism with K-theory is discussed in
An algebraic analogue of h-cobordism:
A construction of algebraic cobordism as a non-$\mathbb{A}^1$-invariant cohomology theory on derived schemes and the resulting Conner-Floyd isomorphism:
Last revised on March 6, 2023 at 11:53:00. See the history of this page for a list of all contributions to it.