# nLab MSU

Contents

### Context

#### Cobordism theory

Concepts of cobordism theory

# Contents

## Properties

### General

We write $\Omega^{SU}_\bullet$ for the bordism ring for stable SU-structure.

###### Proposition

The kernel of the forgetful morphism

$\Omega^{SU}_\bullet \longrightarrow \Omega^{\mathrm{U}}_\bullet$

from the SU-bordism ring to the complex bordism ring, is pure torsion.

###### Proposition

The torsion subgroup of the SU-bordism ring is concentrated in degrees $8k+1$ and $8k+2$, for $k \in \mathbb{N}$.

###### Proposition

Every torsion element in the SU-bordism ring $\Omega^{SU}_\bullet$ has order 2.

###### Proposition

(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 $\geq 4$:

$\Omega^{SU}\big[\tfrac{1}{2}\big] \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2} \big] \big[ \{ y_{2i+4} \}_{i \in \mathbb{N}} \big] \,.$

(due to Novikov 62, review in LLP 17, Thm. 1.2)

### Relation to Calabi-Yau manifolds

We discuss the classes of Calabi-Yau manifolds in the SU-bordism ring. For more see at Calabi-Yau manifolds in SU-bordism theory.

###### Proposition

(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:

$\Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,.$
###### Proposition

(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.

###### Proposition

(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 cobordism homology/cohomology theories and representing Thom spectra

bordism theory$\,M B$ (B-bordism):

## References

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)

Survey:

On its torsion subgroups:

Relation to Calabi-Yau manifolds:

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 03:32:58. See the history of this page for a list of all contributions to it.