geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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:
Since Calabi-Yau manifolds have, in particular SU-structure, they represent classes in SU-bordism theory. In fact, all non-torsion $SU$-bordism classes are represented by products of linear combinations of classes of Calabi-Yau manifolds. Those in degree $4$ are spanned by the K3-surface.
We write $\Omega^{SU}_\bullet$ for the SU-bordism ring.
The kernel of the forgetful morphism
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)
(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, Lemma 1.5, Example 3.1, CLP 19, Theorem 13.5a)
(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.
In particular:
(Calabi-Yau manifolds in complex dim $\leq 4$ generating 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. .
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 November 27, 2020 at 08:48:46. See the history of this page for a list of all contributions to it.