nLab MSU

Idea

The cobordism cohomology theory for special unitary group-structure.

Properties

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

Relation to $MU$

The canonical topological group-inclusions

$$1 \;\subset\; Sp(k) \;\subset\; SU(2k) \;\subset\; U(2k)$$

(trivial group into quaternionic unitary group into special unitary group into unitary group) induce ring spectrum-homomorphism of Thom spectra

$$M Fr \;\longrightarrow\; M Sp \;\longrightarrow\; M SU \;\longrightarrow\; M \mathrm{U}$$

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

(e.g. Conner-Floyd 66, p. 27 (34 of 120))

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.

(CLP 19, Thm. 5.8a)

Proposition

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

(CLP 19, Thm. 5.11a)

Proposition

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

(CLP 19, Thm. 5.8b)

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

(LLP 17, Example 3.1, CLP 19, Theorem 13.5a)

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.

(LLP 17, Theorem 2.4)

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 $\Omega^{SU}_{\leq 8}\big[ \tfrac{1}{2}\big]$.

(CLP 19, Theorem 13.5)

Related concepts

• Calabi-Yau manifold

• generalized Calabi-Yau manifold

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

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

• MFr

• MO, MSO, MSpin, MSpin^c, MSpin^h MString, MFivebrane, M2-Orient, M2-Spin, MNinebrane (see also pin⁻ bordism, pin⁺ bordism, pinᶜ bordism, spin bordism, spinᶜ bordism, spinʰ bordism, string bordism, fivebrane bordism, 2-oriented bordism, 2-spin bordism, ninebrane bordism)

• MU, MSU, MΩΩSU(n)

• MP, MR

• MSp

• MTO, MTSO

• relative bordism theories: MOFr, MUFr, MSUFr

• equivariant bordism theory: equivariant MFr, equivariant MO, equivariant MU

• global equivariant bordism theory: global equivariant mO, global equivariant mU

• algebraic: algebraic cobordism

References

• Pierre Conner, Edwin Floyd, Section 5 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

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:

• Sergei Novikov, p. 218 in: Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (doi:10.1007/978-3-662-10579-5, pdf)

On its torsion subgroups:

• Pierre Conner, Edwin Floyd, Torsion in $SU$-bordism, Memoirs of the American Mathematical Society 1966 (ISBN:978-1-4704-0007-1)

Relation to Calabi-Yau manifolds:

• Ivan Limonchenko, Zhi Lu, Taras Panov, Calabi-Yau hypersurfaces and SU-bordism, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 (arXiv:1712.07350)

On the (failure of) the Conner-Floyd isomorphism for $MSU \to$ KO:

• Serge Ochanine, Modules de SU-bordisme. Applications, Bulletin de la Société Mathématique de France, Tome 115 (1987) , pp. 257-289 (doi:10.24033/bsmf.2078)

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)