e-invariant is Todd class of cobounding (U,fr)-manifold




The e-invariant is, a priori, a homotopy invariant of morphisms in the stable homotopy category, hence in particular of the stable homotopy groups of spheres. Under the identification of the latter, via the Pontrjagin-Thom isomorphism, with the framed bordism ring, the e-invariant turns out to equal the Todd class of any cobounding (U,fr)-manifold (Conner-Floyd 66).



In generalization to how the U-bordism ring Ω 2k U\Omega^U_{2k} is represented by homotopy classes of maps into the Thom spectrum MU, so the (U,fr)-bordism ring Ω 2k U,fr\Omega^{U,fr}_{2k} is represented by maps into the quotient spaces MU 2k/S 2kMU_{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 (U,fr)=π +2k(MU 2k/S 2k),for any2k+2. \Omega^{(U,fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,.

(Conner-Floyd 66, p. 97)


The bordism rings for MU, MUFr and MFr sit in a short exact sequence of the form

(2)0Ω +1 UiΩ +1 U,frΩ fr0, 0 \to \Omega^U_{\bullet+1} \overset{i}{\longrightarrow} \Omega^{U,fr}_{\bullet+1} \overset{\partial}{ \longrightarrow } \Omega^{fr}_\bullet \to 0 \,,

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

(By this Prop. at MUFr.)

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



(Todd class on cobounding (U,fr)-manifolds computes e-invariant)

Evaluation of the Todd class on (U,fr)-manifolds yields rational numbers which are integers on actual UU-manifolds. It follows with the short exact sequence (2) that assigning to FrFr-manifolds the Todd class of any of their cobounding (U,fr)(U,fr)-manifolds yields a well-defined element in Q/Z.

Under the Pontrjagin-Thom isomorphism between the framed bordism ring and the stable homotopy group of spheres π s\pi^s_\bullet, this assignment coincides with the Adams e-invariant in its Q/Z-incarnation:

(3)0 Ω +1 U i Ω +1 U,fr Ω fr π s Td Td e 0 / = /, \array{ 0 \to & \Omega^U_{\bullet+1} & \overset{i}{\longrightarrow} & \Omega^{U,fr}_{\bullet+1} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_\bullet & \simeq & \pi^s_\bullet \\ & \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e_{\mathbb{C}}} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,,

(Conner-Floyd 66, Theorem 16.2)


The first step in the proof of (3) is the observation (Conner-Floyd 66, p. 100-101) that the representing map (1) for a (U,fr)-manifold M 2kM^{2k} cobounding a FrFr-manifold represented by a map ff is given by the following homotopy pasting diagram (see also at Hopf invariantIn generalized cohomology):

homotopy pasting diagram exhibiting cobounding UFr-manifolds
from SS21

From this, Conner-Floyd conclude essentially by considering the following homotopy pasting diagram (the diagrammatic perspective here follows SS21, for more see at Adams e-invariant via unit cofiber theories):

Here for the identification of the Todd class TdTd in the bottom of the diagram we used that the rational Todd class is the Chern character of the Thom class.

That this Todd class equals the Q/Z-valued e-invariant follows now since the latter is the top-degree component of the Chern character of any complex topological K-theory-class V 2nσcV_{2n} \coloneqq \sigma \circ c lifting Σ 2n1K˜(S 2n)\Sigma^{2n} 1 \in \widetilde K(S^{2n}) to C fC_f (this Prop.).


An analogous but finer construction works 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, in variation of (3):

(4)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:


(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).


An alternative formulation via index theory:

Review in:

Last revised on January 15, 2021 at 13:17:45. See the history of this page for a list of all contributions to it.