# nLab MSU

Contents

### Context

#### Cobordism theory

Concepts of cobordism 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:

algebraic:

# Contents

## Properties

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

### Relation to $MU$

The canonical topological group-inclusions

$1 \;\subset\; Sp(k) \;\subset\; SU(2k) \;\subset\; U(2k)$
$M Fr \;\longrightarrow\; M Sp \;\longrightarrow\; M SU \;\longrightarrow\; M \mathrm{U}$

(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

(1)$\Omega^{fr}_{\bullet} \longrightarrow \Omega^{Sp}_{\bullet} \longrightarrow \Omega^{SU}_{\bullet} \longrightarrow \Omega^{U}_{\bullet}$
###### Proposition

The kernel of the forgetful morphism (1)

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

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:

(2)$\Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big\langle -[K3] \big\rangle \,.$
###### Corollary

(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

$\array{ \Omega^{SU}_4 &\hookrightarrow& \Omega^{\mathrm{U}}_4 \\ [K3] &\mapsto& [K3] \,. }$

(This is vaguely indicated in Novikov 86, p. 216 (218 of 321).)

###### Proof

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. ).

###### 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 bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

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

relative bordism theories:

algebraic:

## References

On the SU-bordism ring structure away from 2:

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 10:09:19. See the history of this page for a list of all contributions to it.