nLab MSUFr

Idea

In joint generalization of the cobordism cohomology theories MSU and MFr of closed $SU$-manifolds and of $Fr$-manifolds, respectively, an $(SU,fr)$-manifold (Conner-Floyd 66, Section 16, p. 103 onwards) is a compact manifold with boundary equipped with special unitary group-tangential structure on its stable tangent bundle and equipped with a trivialization (stable framing) of that over the boundary.

The corresponding bordism classes form a bordism ring denoted $\Omega^{SU,fr}_\bullet$.

Properties

Representing spectrum

In generalization to how $\Omega^{SU}_{2k}$ is represented by homotopy classes of maps into the Thom spectrum MSU, so $\Omega^{SU,fr}_{2k}$ is represented by maps into the quotient spaces $MSU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{U(k)} E U(k) ) = MU_{2k}$ the canonical inclusion):

(1)$$\Omega^{(SU,fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MSU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,.$$

(Conner-Floyd 66, p. 103)

Relation to $MSU$ and $MFr$

In every eighth degree, the bordism rings for MSU, $MSUFr$ and MFr sit in a short exact sequence of the form (Conner-Floyd 66, p. 104):

(2)$$0 \to \Omega^{SU}_{8\bullet+4} \overset{i}{ \longrightarrow } \Omega^{SU,fr}_{8\bullet+4} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{8\bullet + 3} \to 0 \,,$$

where $i$ is the evident inclusion, while $\partial$ is restriction to the boundary.

In particular, this means that $\partial$ is surjective, hence that every $Fr$-manifold of dimension $8k + 3$ is the boundary of a $(U,fr)$-manifold.

Relation to Todd classes and the $e_{\mathbb{R}}$-invariant

In refinement of how the complex e-invariant is the Todd class of cobounding (U,fr)-manifolds we have for special unitary group-structure instead of unitary group-structure and in dimensions $8\bullet + 4$:

Since on $(8 \bullet + 4)$-dimensional $SU$-manifolds the Todd class is divisible by 2 Conner-Floyd 66, Prop. 16.4, we have (Conner-Floyd 66, p. 104) the following short exact sequence of MSUFr-bordism rings:

(3)$$\array{ 0 \to & \Omega^{SU}_{8\bullet+4} & \overset{i}{\longrightarrow} & \Omega^{SU,fr}_{8\bullet+4} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_{8\bullet + 3} & \simeq & \pi^s_{8\bullet+3} \\ & \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}} && \big\downarrow{}^{\tfrac{1}{2}\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e_{\mathbb{R}}} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,.$$

This produces $e_{\mathbb{R}}$, the Adams e-invariant with respect to KO-theory instead of KU (Adams 66, p. 39), which, in degrees $8k + 3$, is indeed half of the e-invariant $e_{\mathbb{C}}$ for $KU$ (by Adams 66, Prop. 7.14).

In fact, for $k = 0$ we have:

Proposition

(Adams 66, Example 7.17 and p. 46)

In degree 3, the KO-theoretic e-invariant $e_{\mathbb{R}}$ takes the value $\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z}$ on the quaternionic Hopf fibration $S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4$ and hence reflects the full third stable homotopy group of spheres:

$$\array{ \pi^s_3 & \underoverset{ \simeq }{ e_{\mathbb{R}} }{ \;\;\longrightarrow\;\; } & \mathbb{Z}/24 & \subset & \mathbb{Q}/\mathbb{Z} \\ [h_{\mathbb{H}}] &&\mapsto&& \left[\tfrac{1}{24}\right] }$$

while $e_{\mathbb{C}}$ sees only “half” of it (by Adams 66, Prop. 7.14).

Related concepts

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, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

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

The concept of $(SU,fr)$-bordism theory and its relation to the e-invariant originates with:

• Pierre Conner, Edwin Floyd, Section 16 from p. 103 onwards, in: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)