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:
The cobordism cohomology theory for special unitary group-structure.
We write $\Omega^{SU}_\bullet$ for the bordism ring for stable SU-structure.
The canonical topological group-inclusions
(trivial group into quaternionic unitary group into special unitary group into unitary group) induce ring spectrum-homomorphism of Thom spectra
(from MFr to MSp to MSU to MU)
and hence corresponding multiplicative cohomology theory-homomorphisms of cobordism cohomology theories, so in particular ring homomorphisms of bordism rings
(e.g. Conner-Floyd 66, p. 27 (34 of 120))
The kernel of the forgetful morphism (1)
from the SU-bordism ring to the complex bordism ring, is pure torsion.
The torsion subgroup of the SU-bordism ring is concentrated in degrees $8k+1$ and $8k+2$, for $k \in \mathbb{N}$.
Every torsion element in the SU-bordism ring $\Omega^{SU}_\bullet$ has order 2.
(SU-bordism ring away from 2 is polynomial algebra)
The SU-bordism ring with 2 inverted is the polynomial algebra over $\mathbb{Z}\big[\tfrac{1}{2}\big]$ on one generator in every even degree $\geq 4$:
(due to Novikov 62, review in LLP 17, Thm. 1.2)
We discuss the classes of Calabi-Yau manifolds in the SU-bordism ring. For more see at Calabi-Yau manifolds in SU-bordism theory.
(K3-surface spans SU-bordism ring in degree 4)
The degree-4 generator $y_4 \in \Omega^{SU}_4$ in the SU-bordism ring (Prop. ) is represented by minus the class of any (non-torus) K3-surface:
(LLP 17, Example 3.1, CLP 19, Theorem 13.5a)
(K3-surface represents non-trivial element in U-bordism ring)
The image in the MU-cobordism ring of the class of the K3-surface $[K3] \in \Omega^{SU}_4$ (2) under the canonical morphism $\Omega^{SU}_4 \to \Omega^{\mathrm{U}}_4$ (1) is non-trivial.
In fact, the canonical morphism is an injection in this degree
(This is vaguely indicated in Novikov 86, p. 216 (218 of 321).)
By Prop. the kernel of the map to $\Omega^{\mathrm{U}}_4$ is torsion, but by Prop. $[K3]$ represents a non-torsion element. Since it is in fact a non-torsion generator, the kernel vanishes (as also implied by Prop. ).
(Calabi-Yau manifolds generate the SU-bordism ring away from 2)
The SU-bordism ring away from 2 is multiplicatively generated by Calabi-Yau manifolds.
(Calabi-Yau manifolds in complex dim $\leq 4$ span the SU-bordism ring in $deg \leq 8$ away from 2)
There are Calabi-Yau manifolds of complex dimension $3$ and $4$ whose whose SU-bordism classes equal the generators $\pm y_6$ and $\pm y_8$ in Prop. .
Together with the K3 surface representing $- y_4$ (Prop. ), this means that CYs span $\Omega^{SU}_{\leq 8}\big[ \tfrac{1}{2}\big]$.
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:
On the SU-bordism ring structure away from 2:
Sergei Novikov, Homotopy properties of Thom complexes, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (pdf, pdf)
Robert Stong, Chapter X of: Notes on Cobordism theory, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016, pdf)
Survey:
On its torsion subgroups:
Relation to Calabi-Yau manifolds:
On the (failure of) the Conner-Floyd isomorphism for $MSU \to$ KO:
Survey:
Georgy Chernykh, Ivan Limonchenko, Taras Panov, $SU$-bordism: structure results and geometric representatives, Russian Math. Surveys 74 (2019), no. 3, 461-524 (arXiv:1903.07178)
Taras Panov, A geometric view on $SU$-bordism, talk at Moscow State University 2020 (webpage, pdf, pdf)
Last revised on February 18, 2021 at 15:09:19. See the history of this page for a list of all contributions to it.