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MSU

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Contents

Idea

The cobordism cohomology theory for special unitary group-structure.

Properties

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

Relation to MUMU

The canonical topological group-inclusions

1Sp(k)SU(2k)U(2k) 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

MFrMSpMSUMU 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)Ω frΩ SpΩ SUΩ U \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)

Ω SUΩ U \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+18k+1 and 8k+28k+2, for kk \in \mathbb{N}.

(CLP 19, Thm. 5.11a)

Proposition

Every torsion element in the SU-bordism ring Ω SU\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 [12]\mathbb{Z}\big[\tfrac{1}{2}\big] on one generator in every even degree 4\geq 4:

Ω SU[12][12][{y 2i+4} i]. \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 4Ω 4 SUy_4 \in \Omega^{SU}_4 in the SU-bordism ring (Prop. ) is represented by minus the class of any (non-torus) K3-surface:

(2)Ω 4 SU[K3]. \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]Ω 4 SU[K3] \in \Omega^{SU}_4 (2) under the canonical morphism Ω 4 SUΩ 4 U\Omega^{SU}_4 \to \Omega^{\mathrm{U}}_4 (1) is non-trivial.

In fact, the canonical morphism is an injection in this degree

Ω 4 SU Ω 4 U [K3] [K3]. \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 Ω 4 U\Omega^{\mathrm{U}}_4 is torsion, but by Prop. [K3][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 4\leq 4 span the SU-bordism ring in deg8deg \leq 8 away from 2)

There are Calabi-Yau manifolds of complex dimension 33 and 44 whose whose SU-bordism classes equal the generators ±y 6\pm y_6 and ±y 8\pm y_8 in Prop. .

Together with the K3 surface representing y 4- y_4 (Prop. ), this means that CYs span Ω 8 SU[12]\Omega^{SU}_{\leq 8}\big[ \tfrac{1}{2}\big].

(CLP 19, Theorem 13.5)


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:

equivariant bordism theory:

global equivariant bordism theory:

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 MSUMSU \to KO:

Survey:

Last revised on February 18, 2021 at 10:09:19. See the history of this page for a list of all contributions to it.