nLab
MSUFr

Contents

Contents

Idea

In joint generalization of the cobordism cohomology theories MSU and MFr of closed SUSU-manifolds and of FrFr-manifolds, respectively, an (SU,fr)(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 Ω SU,fr\Omega^{SU,fr}_\bullet.

Properties

Representing spectrum

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

(1)Ω (SU,fr)=π +2k(MSU 2k/S 2k),for any2k+2. \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 MSUMSU and MFrMFr

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

(2)0Ω 8+4 SUiΩ 8+4 SU,frΩ 8+3 fr0, 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 ii is the evident inclusion, while \partial is restriction to the boundary.

In particular, this means that \partial is surjective, hence that every FrFr-manifold of dimension 8k+38k + 3 is the boundary of a (U,fr)(U,fr)-manifold.

Relation to Todd classes and the e 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+48\bullet + 4:

Since on (8+4)(8 \bullet + 4)-dimensional SUSU-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)0 Ω 8+4 SU i Ω 8+4 SU,fr Ω 8+3 fr π 8+3 s 12Td 12Td e 0 / = /. \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 e_{\mathbb{R}}, the Adams e-invariant with respect to KO-theory instead of KU (Adams 66, p. 39), which, in degrees 8k+38k + 3, is indeed half of the e-invariant e e_{\mathbb{C}} for KUKU (by Adams 66, Prop. 7.14).

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

Proposition

(Adams 66, Example 7.17 and p. 46)

In degree 3, the KO-theoretic e-invariant e e_{\mathbb{R}} takes the value [124]/\left[\tfrac{1}{24}\right] \in \mathbb{Q}/\mathbb{Z} on the quaternionic Hopf fibration S 7h S 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4 and hence reflects the full third stable homotopy group of spheres:

π 3 s e /24 / [h ] [124] \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 e_{\mathbb{C}} sees only “half” of it (by Adams 66, Prop. 7.14).

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

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

Created on December 18, 2020 at 05:54:42. See the history of this page for a list of all contributions to it.