nLab
Adams e-invariant

Contents

Contents

Idea

The ee-invariant (Adams 66, Sections 3,7, short for “extension invariant”, see Def. below) is the second in a sequence of homotopy invariants of “stable maps”, i.e. of morphisms in the stable homotopy category (in particular: of stable homotopy groups of spheres), being elements of Ext-groups between the homology groups/cohomology groups of the domain and codomain of the map, with respect to some suitable choice of Whitehead-generalized cohomology theory EE.

The previous invariant in the sequence is the d-invariant, the next is the f-invariant. These are the elements that appear in the first lines on the second page of the EE-Adams spectral sequence for [X,Y] [X,Y]_\bullet.

Definition

Let XfYX \overset{f}{\longrightarrow} Y be morphism in the stable homotopy category out of a finite spectrum XX (for instance the image under suspension Σ \Sigma^\infty of a morphism in the classical homotopy category of pointed homotopy types out of a finite CW-complex).

Let EE be a multiplicative cohomology theory, such that the d-invariant of ff in EE vanishes, hence such that pullback f *:E (Y)E (X)f^\ast \;\colon\; E^\bullet(Y) \to E^\bullet(X) in EE-cohomology is the zero morphism.

The archetypical example is f:S 2n1S 2nf \;\colon\; S^{2n-1} \to S^{2n} a map out of an odd-dimensional sphere and E=KUE = KU complex topological K-theory.


As an extension of generalized Adams-operation modules

Definition

(e-invariant as extension class in E-cohomology)

Writing C fC_f for the homotopy cofiber of ff

XfYC fΣX, \cdots \to X \overset{f}{\longrightarrow} Y \overset{}{\longrightarrow} C_f \overset{}{\longrightarrow} \Sigma X \to \cdots \,,

this implies that the long exact sequence in cohomology corresponding to the pair (Y,C f)(Y, C_f) truncates to a short exact sequence of the form

(1)0E (ΣX)E (C f)E (Y)0. 0 \to E^\bullet(\Sigma X) \overset{}{\longrightarrow} E^\bullet(C_f) \overset{}{\longrightarrow} E^\bullet(Y) \to 0 \,.

This is hence an extension of E (Y)E^\bullet(Y) by E (ΣX)E^\bullet(\Sigma X) in any category in which f *f^\ast is a homomorphism, for instance that of modules over the E-Steenrod algebra. For the case of E=KUE = KU take the category of graded abelian groups equipped with Adams operations.

Thus this short exact sequence defines an element in the Ext group formed in this category

(2)e(f)Ext 1(E (Y),E (ΣX)) e(f) \;\in\; Ext^1\big( E^\bullet(Y), \, E^\bullet(\Sigma X) \big)

and this is the e-invariant of ff seen in EE-theory.

(Adams 66, Section 3, p. 27, review includes BL 09, Sec. 2).


For E=KUE = KU

As an extension of K-groups with Adams operations

We discuss this in more detail for the case of complex topological K-theory E=KUE = KU, and for f:S 2(n+)1S 2nf \;\colon\; S^{2(n + \bullet)-1} \to S^{2n} a map between spheres; and we show how the resulting extension is characterized by a single rational number modulo the integers, this being the e-invariant in the form of a Q/Z-valued character

e :π s/ e_{\mathbb{C}} \;\colon\; \pi^s_{\bullet} \overset{\;\;\;\;\;}{\longrightarrow} \mathbb{Q}/\mathbb{Z}

on stable homotopy groups of spheres (Def. below).

We follow Hopkins-Mathew 12, Lecture 11.


Abelian groups with Adams operations

Definition

(abelian group with Adams operations)

We say that an abelian group with Adams operation (A,{ψ A k} k)\big(A,\{\psi_A^k\}_{k \in \mathbb{N}}\big) is an abelian group AA equipped with an action of the multiplicative monoid (,)(\mathbb{N}, \cdot) of natural numbers, hence equipped with group homomorphisms

ψ A k:AA,for allk \psi_A^k \;\colon\; A \to A \,, \;\;\;\;\;\;\; \text{for all} \; k \in \mathbb{N}

such that

(3)ψ A k 1ψ A k 2=ψ A k 1k 2,for allk 1,k 2. \psi_A^{k_1} \circ \psi_A^{k_2} \;=\; \psi_A^{ k_1 \cdot k_2 } \,, \;\;\;\;\;\;\;\;\; \text{for all} \; k_1, k_2 \,\in\, \mathbb{N} \,.

Moreover, for (A,{ψ A k} k),(A,{ψ A k} k)\big(A,\{\psi_A^k\}_{k \in \mathbb{N}}\big), \, \big(A',\{\psi_{A'}^k\}_{k \in \mathbb{N}}\big) two abelian groups with Adams operations, a homomorphism between them is a group homomorphism (a linear map) ϕ:AA\phi \;\colon\; A \to A' that respects these operations, hence such that the following squares commute:

(4)A ϕ A ψ A k ψ A k A ϕ A,for allk. \array{ A &\overset{\;\;\;\phi\;\;\;}{\longrightarrow}& A' \\ {}^{\mathllap{ \psi^k_A }} \big\downarrow && \big\downarrow {}^{\mathrlap{\psi_{A'}^k}} \\ A &\overset{\;\;\;\phi\;\;\;}{\longrightarrow}& A' \,, } \;\;\;\;\;\;\;\; \text{for all} \; k \in \mathbb{N} \,.

This makes an abelian category which we denoted Ab AdamsAb_{Adams}, canonically equipped with a forgetful functor to Ab:

(5)Ab Adams Ab (A,{ψ A k} k) A. \array{ Ab_{Adams} &\overset{\;\;\;\;\;\;}{\longrightarrow}& Ab \\ \big( A, \, \{\psi^k_A\}_{k \in \mathbb{N}} \big) &\mapsto& A \,. }
Example

For nn \in \mathbb{N}, the additive abelian group of integers \mathbb{Z} becomes an abelian group with Adams operations

(n)(,{ψ kk}), \mathbb{Z}(n) \;\coloneqq\; \big( \mathbb{Z},\, \{\psi^k_{\mathbb{Z}}_k\} \big) \,,

in the sense of Def. , by setting

ψ k r k nr. \array{ \mathbb{Z} & \overset{\;\;\; \psi^k\;\;\;}{\longrightarrow} & \mathbb{Z} \\ r &\mapsto& k^n \cdot r \,. }
Example

(Adams operations on complex topological K-theory groups)

For XX a compact pointed topological space, the complex topological K-theory group K(X)K(X) becomes an abelian group with Adams operations in the sense of Def. , via the actual Adams operations,

K(X),K˜(X)Ab Adams K(X), \widetilde K(X) \;\in\; Ab_{Adams}

and hence so does the reduced K-theory K˜(X)\widetilde K(X):

K˜(X) ker(i *) K(X) i * K(*) ψ K˜(X) k ψ K(X) k ψ K(*) k K˜(X) ker(i *) K(X) i * K(*), \array{ \widetilde K(X) &\overset{\;\;ker(i^\ast)\;\;}{\longrightarrow}& K(X) &\overset{\;i^\ast\;}{\longrightarrow}& K(\ast) \\ \big\downarrow {}^{\mathrlap{\psi^k_{\widetilde K(X)}}} && \big\downarrow {}^{\mathrlap{\psi^k_{K(X)}}} && \big\downarrow {}^{\mathrlap{\psi^k_{K(\ast)}}} \\ \widetilde K(X) &\underset{\;\;ker(i^\ast)\;\;}{\longrightarrow}& K(X) &\underset{\;i^\ast\;}{\longrightarrow}& K(\ast) \,, }

Moreover, for each (pointed) continuous function XfYX \overset{f}{\longrightarrow} Y the corresponding pullback in cohomology respects the Adams operations and hence yields a homomorphism (4):

(6)K˜(X)f *K˜(Y),Ab Adams. \widetilde K(X) \overset{\;\;f^\ast\;\;}{\longrightarrow} \widetilde K(Y) \,, \;\;\;\; \in \; Ab_{Adams} \,.
Proposition

(Adams operations on complex topological K-theory of n-spheres)

For nn \in \mathbb{N}, the Adams operations on the reduced K-theory (Example ) of the 2n-sphere are given by:

K˜(S 2n) ψ k K˜(S 2n) V k nV \array{ \widetilde K \big( S^{2n} \big) & \overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow} & \widetilde K \big( S^{2n} \big) \\ V &\mapsto& k^n \cdot V }

and hence are isomorphic in Ab AdamsAb_{Adams} (Def. ) to the objects from Example :

K˜(S 2n)(n)Ab Adams. \widetilde K \big( S^{2n} \big) \;\simeq\; \mathbb{Z}(n) \;\;\;\; \in \; Ab_{Adams} \,.

The Adams e e_{\mathbb{C}}-invariant in /\mathbb{Q}/\mathbb{Z}

Example

(the defining short exact sequence in complex topological K-theory)

For n,nn, n' \,\in\, \mathbb{N} let

f:S 2(n+n)1S 2n f \;\colon\; S^{2(n + n') - 1} \longrightarrow S^{2n}

be a continuous function between spheres (representing its image in the classical homotopy category and in fact in the stable homotopy category, which is all that its image in a Whitehead-generalized cohomology theory such as complex topological K-theory depends on), such that the d-invariant of ff vanishes under complex topological K-theory E=KUE = KU, hence such that the pullback f *f^\ast is the zero map in K-theory:

d(f)f *=0:K˜(S 2n)K˜(S 2(n+n)1). d(f) \;\coloneqq\; f^\ast \;=\; 0 \;\;\;\colon\; \widetilde K\big(S^{2n}\big) \longrightarrow \widetilde K\big( S^{2(n + n') - 1} \big) \,.

Writing C fC_f for the homotopy type of the homotopy cofiber/attaching space of ff:

(7)S 2(n+n)1 * f (hpo) S 2n i 2n C f (hpo) p 2(n+n) * S 2(n+n)S 2(n+n)1 D 2(n+n) f (po) S 2n i 2n C f (po) p 2(n+n) * S 2(n+n) \array{ S^{2(n + n') - 1} &\longrightarrow& \ast \\ {}^{\mathllap{f}} \big\downarrow & {}^{_{(hpo)}} & \big\downarrow \\ S^{2n} &\underset{i_{2n}}{\longrightarrow}& C_f \\ {}^{} \big\downarrow & {}^{_{(hpo)}} & \big\downarrow {}^{\mathrlap{ p_{2(n+n')} }} \\ \ast &\longrightarrow& S^{2(n + n')} } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ S^{2(n + n') - 1} &\longrightarrow& D^{2(n + n')} \\ {}^{\mathllap{f}} \big\downarrow & {}^{_{(po)}} & \big\downarrow \\ S^{2n} &\underset{i_{2n}}{\longrightarrow}& C_f \\ {}^{} \big\downarrow & {}^{_{(po)}} & \big\downarrow {}^{\mathrlap{ p_{2(n+n')} }} \\ \ast &\longrightarrow& S^{2(n + n')} }

this implies that the long exact sequence in cohomology induced by the CW-pair S 2nC fS^{2n} \hookrightarrow C_f truncates to the short exact sequence (1), here regarded, by Example , in the abelian category Ab AdamsAb_{Adams} of abelian groups with Adams operations (Def. ):

(8)0 K˜(S 2(n+n)) p 2(n+n) * K˜(C f) i 2n * K˜(S 2n) 0 = 0 (n+n) K˜(C f) (n) 0Ab Adams, \array{ 0 &\to& \widetilde K\big( S^{2(n + n')} \big) &\overset{\;\;\; p_{2(n+n')}^\ast \;\;\;}{\longrightarrow}& \widetilde K\big( C_f \big) &\overset{\;\;\; i_{2n}^\ast \;\;\;}{\longrightarrow}& \widetilde K\big( S^{2n} \big) &\to& 0 \\ && \big\downarrow {}^{\mathrlap{\simeq}} && \big\downarrow {}^{\mathrlap{=}} && \big\downarrow {}^{\mathrlap{\simeq}} \\ 0 &\to& \mathbb{Z}(n + n') &\underset{\;\;\; \;\;\;\; \;\;\;}{\longrightarrow}& \widetilde K\big( C_f \big) &\underset{\;\;\; \;\;\;\; \;\;\;}{\longrightarrow}& \mathbb{Z}(n) &\to& 0 } \;\;\;\;\;\;\;\; \in \; Ab_{Adams} \,,

where in the second line we have identified the outer groups via Prop. .

By definition of Ext-groups, the isomorphism class of this short exact sequence with the outer groups fixed is an element

(9)e (f)Ext Ab Adams 1((n),(n+n)). e_{\mathbb{C}}(f) \;\;\in\;\; Ext^1_{Ab_{Adams}} \big( \mathbb{Z}(n), \, \mathbb{Z}(n + n') \big) \,.

This is the Adams e-invariant of ff as seen in complex topological K-theory; in specialization of (2).

More concretely, it turns out that the extension (9) is completely characterized by a single rational number modulo this integers. This we discuss next (Def. below).

Remark

(extension after forgetting the Adams module structure is trivial)

After forgetting the action of the Adams operations via (5), the sequence (8) is still a short exact sequence, now of plain abelian groups. However, since Ext Ab 1(,)=0Ext^1_{Ab}(\mathbb{Z},-) = 0 (this Prop.), it is necessarily trivial as an extension, showing that the underlying abelian cohomlogy group of the cofiber space is just

(10) K˜(S 2n)K˜(S 2(n+n)) K˜(C f) (1,1) (Σ 2n1,Σ 2(n+n)1) (V 2n,V 2(n+n))Ab, \array{ \mathbb{Z} \oplus \mathbb{Z} & \simeq & \widetilde K \big( S^{2n} \big) \,\oplus\, \widetilde K \big( S^{2(n + n')} \big) & \simeq & \widetilde K \big( C_f \big) \\ \big( 1, \, 1 \big) &\mapsto& \big( \Sigma^{2n} 1, \, \Sigma^{2(n + n')} 1 \big) &\mapsto& \big( V_{2n}, \, V_{2(n+n')} \big) } \;\; \;\in\; Ab \,,

where V 2(n+n)p 2(n+n) *Σ 2(n+n)1V_{2(n + n')} \;\coloneqq\; p_{2(n+n')}^\ast \Sigma^{2(n+n')} 1 and where V 2nV_{2n} is a choice of lift of Σ 2n1\Sigma^{2n} 1 through i 2n *i_{2n}^\ast in the short exact sequence (8):

(11)0 K˜(S 2(n+n)) p 2(n+n) * K˜(C f) i 2n * K˜(S 2n) 0 Σ 2(n+n)1 V 2(n+n) V 2n Σ 2n1. \array{ 0 \to & \widetilde K\big( S^{2(n+n')} \big) & \overset{p^\ast_{2(n+n')}}{\longrightarrow} & \widetilde K\big( C_f \big) & \overset{i^\ast_{2n}}{\longrightarrow} & \widetilde K\big( S^{2n} \big) & \to 0 \\ & \Sigma^{2(n+n')} 1 &\mapsto& V_{2(n+n')} \\ & && V_{2n} &\mapsto& \Sigma^{2n} 1 \,. }

Notice that the isomorphism (10) depends on a choice of splitting (11) of the short exact sequence (8) in Ab: any two choices V 2nV_{2n}, V 2nV'_{2n} differ by a multiple ss \in \mathbb{Z} of the generator V 2(n+n)V_{2(n + n')}:

(12)V 2n=V 2n+sV 2(n+n). V'_{2n} \;=\; V_{2n} + s \cdot V_{2(n + n')} \,.

Conversely, this means that all the information in the extension (8) is in how the Adams operations act on the K-theory of the cofiber space:

Definition

(e-invariant in complex topological K-theory as rational number modulo integers)

Given a map f:S 2(n+n)1S 2nf \;\colon\; S^{2(n + n') - 1} \longrightarrow S^{2n} with vanishing KUKU-d-invariant, as in Example , we have by Remark that the Adams operations (Example ) on the cofiber space C fC_f (7) must be of the following form (Adams 66, Prop. 7.5, (7.10) (9.2)):

(13)K˜(C f) ψ k K˜(C f) V 2n k nV 2n+μ k(f)V 2(n+n) V 2(n+n) k n+nV 2(n+n). \array{ \widetilde K\big( C_f\big) &\overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow}& \widetilde K\big( C_f\big) \\ V_{2n} &\mapsto& k^n \cdot V_{2n} \;+\; {\color{blue} \mu_k(f) } \cdot V_{2(n+ n')} \\ V_{2(n + n')} &\mapsto& k^{n + n'} \cdot V_{2(n + n')} \;\;\;\;\;\;\;\; \,. }

Namely, the first summands on the right are constrained to be as shown, by Prop. and using that pullback in cohomology i 2n *i^\ast_{2n}, p 2(n+n) *p^\ast_{2(n + n')}(6) respects the Adams operations (Example ); while the second summand, which vanishes under i 2n *i^\ast_{2n}, must be some multiple

μ k(f) \mu_k(f) \;\in\; \mathbb{Z}

of the only other generator V 2(n+n)V_{2(n + n')} (10). This μ\mu is the only part of the data that is not completely fixed by the Adams module structure, and which may depend on the map ff.

We say that the Adams e-invariant of ff is this multiple μ\mu, normalized as a rational number as follows, and then regarded modulo addition of integers as an element in Q/Z:

(14)e (f)[μ k(f)k n(k n1)]/. e_{\mathbb{C}}(f) \;\coloneqq\; \left[ \frac{ \mu_k(f) }{ k^{n} \big( k^{n'} - 1 \big) } \right] \;\; \in \;\; \mathbb{Q}/\mathbb{Z} \,.
Proposition

(e-invariant as rational number modulo integers is well defined)

The e-invariant e (f)e_{\mathbb{C}}(f) (14) from Def. is well-defined, in that it is

  1. independent of the choice of splitting Σ 2n1V 2n\Sigma^{2n} 1 \,\mapsto\, V_{2n} in (10);

  2. independent of the choice of kk on the right of (14).

Proof

On 1. Under a different choice of splitting, V 2nV_{2n} changes to (12)

V 2n=V 2n+sV 2(n+n) V'_{2n} \;=\; V_{2n} + s \cdot V_{2(n + n')}

for some ss \in \mathbb{Z}. By inspection of (13) this implies that μ k(f)\mu_k(f) changes to

μ k(f)=μ k(f)+s(k n+nk n); \mu'_k(f) \;=\; \mu_k(f) + s \cdot \big( k^{n + n'} - k^n \big) \,;

and so in (14) we have

e (f)=[μ k(f)k n(k n1)]=[μ k(f)k n(k n1)+s]=[μ k(f)k n(k n1)]=e (f). e'_{\mathbb{C}}(f) \;=\; \left[ \frac{ \mu'_k(f) }{ k^n(k^{n'} -1 ) } \right] \;=\; \left[ \frac{ \mu_k(f) }{ k^n(k^{n'} -1 ) } + s \right] \;=\; \left[ \frac{ \mu_k(f) }{ k^n(k^{n'} -1 ) } \right] \;=\; e_{\mathbb{C}}(f) \,.

On 2. Use the commutativity (3) of the Adams operation together with the formula (13) to find for any k 1,k 2k_1, k_2 \,\in\, \mathbb{N}:

=ψ k 1ψ k 2(V 2n) =ψ k 2ψ k 1(V 2n) =ψ k 1(k 2 nV 2n+μ k 2(f)V 2(n+n)) =ψ k 2(k 1 nV 2n+μ k 1(f)V 2(n+n)) =(k 1k 2) nV 2n+(k 2 nμ k 1(f)+k 1 n+nμ k 2(f))V 2(n+n) =(k 1k 2) nV 2n+(k 1 nμ k 2(f)+k 2 n+nμ k 1(f))V 2(n+n) =k 1 n(k 1 n1)μ k 2(f) =k 2 n(k 2 n1)μ k 1(f) =μ k 1(f)k 1 n(k 1 n1) =μ k 2(f)k 2 n(k 2 n1) \begin{aligned} & \phantom{\;=\;\;} \psi^{k_1} \circ \psi^{k_2} \big( V_{2n} \big) \\ & \;=\; \psi^{k_2} \circ \psi^{k_1} \big( V_{2n} \big) \\ \\ \Leftrightarrow \;\;\; & \phantom{\;=\;\;} \psi^{k_1} \big( k_2^n \cdot V_{2n} + \mu_{k_2}(f) \cdot V_{2(n + n')} \big) \\ & \;=\; \psi^{k_2} \big( k_1^n \cdot V_{2n} + \mu_{k_1}(f) \cdot V_{2(n + n')} \big) \\ \\ \Leftrightarrow \;\;\; & \phantom{\;=\;\;} (k_1 k_2)^n \cdot V_{2n} + \Big( k_2^n \mu_{k_1}(f) + k_1^{n + n'} \mu_{k_2}(f) \Big) \cdot V_{2(n + n')} \\ & \;=\; (k_1 k_2)^n \cdot V_{2n} + \Big( k_1^n \mu_{k_2}(f) + k_2^{n + n'} \mu_{k_1}(f) \Big) \cdot V_{2(n + n')} \\ \\ \Leftrightarrow \;\;\; & \phantom{\;=\;\;} k_1^n \big( k_1^{n'} - 1 \big) \mu_{k_2}(f) \\ & \;=\; k_2^n \big( k_2^{n'} - 1 \big) \mu_{k_1}(f) \\ \\ \Leftrightarrow \;\;\; & \phantom{\;=\;\;} \frac{ \mu_{k_1}(f) }{ k_1^n \big( k_1^{n'} - 1 \big) } \\ & \;=\; \frac{ \mu_{k_2}(f) }{ k_2^n \big( k_2^{n'} - 1 \big) } \end{aligned}


As the top degree Chern character on cofiber space

Proposition

(Q/Z-valued e-invariant is top-degree coefficient of Chern character on cofiber space)

In the situation of Example , with

f:S 2(n+n)1S 2n f \;\colon\; S^{2(n+n')-1 } \longrightarrow S^{2n}

a map between spheres, and with V 2nK˜(C f)V_{2n} \,\in\, \widetilde K\big( C_f \big) any lift (11) of Σ 2n1K˜(S 2n)\Sigma^{2 n} 1 \,\in\, \widetilde K \big( S^{2n} \big) to its homotopy cofiber space, we have that the e-invariant e (f)e_{\mathbb{C}}(f) (Def. ) is equivalently the evaluation modulo integers of the Chern character ch(V 2n)H ev(C f;)ch(V_{2n}) \,\in\, H^{ev}\big( C_f;\, \mathbb{Q} \big) on the fundamental class of the cofiber space:

exp(2πi C fch(V 2n))=exp(2πie (f))U(1). \exp \left( 2 \pi \mathrm{i} \int_{C_f} ch\big( V_{2n} \big) \right) \;=\; \exp \left( 2 \pi \mathrm{i} \, { \color{blue} e_{\mathbb{C}}(f) } \right) \;\;\; \in \mathrm{U}(1) \,.

Adams 66, Prop. 7.5

Proof

By (13) we have, now in matrix calculus-notation:

ψ k:[V 2n V 2(n+n)][k n e(f)k n(k n1) 0 k n+n][V 2n V 2(n+n)]K˜(C f). \psi^k \;\colon\; \left[ \array{ V_{2 n } \\ V_{2(n + n')} } \right] \;\;\; \mapsto \;\;\; \left[ \array{ k^n & e(f) \, k^n (k^{n'} - 1) \\ 0 & k^{n + n'} } \right] \cdot \left[ \array{ V_{2 n } \\ V_{2(n + n')} } \right] \;\;\;\;\; \in \; \widetilde K\big( C_f \big) \,.

This matrix has two eigenvectors over the rational numbers (in general). Therefore we now consider the image of these K-theory classes under the Chern character map

ch:K˜(C f)H ev(C f;). ch \;\colon\; \widetilde K\big( C_f \big) \longrightarrow H^{ev}\big( C_f;\, \mathbb{Q}\big) \,.

Since the Adams operations are compatible with the Chern character, we then have the following eigenvectors of the Adams operations under chch:

(15)ψ H k:{ch(V 2n)e(f)ch(V 2(n+n)) k n (ch(V 2n)e(f)ch(V 2(n+n))) ch(V 2(n+n)) k n+n ch(V 2(n+n))H ev(C f;)/H 2(n+n)(C f;). \psi^k_H \;\;\colon\;\; \left\{ \array{ ch \big( V_{2 n} \big) - e(f) \cdot ch \big( V_{2(n + n')} \big) & \mapsto & k^{n } & \cdot & \big( ch ( V_{2 n} ) - e(f) \cdot ch ( V_{2(n + n')} ) \big) \\ ch ( V_{2(n+n')} ) & \mapsto & k^{n + n'} & \cdot & ch ( V_{2 (n + n')} ) } \right. \;\;\;\; \in \; H^{ev}\big( C_f; \, \mathbb{Q} \big)/H^{2(n+n')}\big( C_f; \, \mathbb{Z} \big) \,.

Here, since V 2nV_{2n} is well defined modulo addition (12) of integral multiples of V 2(n+n)V_{2(n+n')}, and since

(16)ch(V 2(n+n))=1H 2(n+n)(C f;), ch\big( V_{2(n+n')} \big) \;=\; 1 \,\in\, \mathbb{Z} \,\simeq\, H^{2(n+n')}\big( C_f;\, \mathbb{Z} \big) \,,

this expression (15) is well-defined in ordinary rational cohomology in even degrees modulo integral cohomology in top degree.

But since the eigenvectors of ψ H k\psi^k_H to eigenvalue k rk^r are precisely the ordinary cohomology classes in homogeneous degree H 2r(C f;)H ev(C f;)H^{2r}\big( C_f;\, \mathbb{Q} \big) \,\subset\, H^{ev}\big( C_f;\, \mathbb{Q} \big) (see there), this means that

ch(V 2n)=ch(V 2n)e(f)ch(V 2(n+n))H 2n(C f;)+e(f)ch(V 2(n+n)).H 2(n+n)(C f;/) ch \big( V_{2n} \big) \;\;=\;\; \underset{ \in \; H^{2n}\big( C_f; \, \mathbb{Q} \big) }{ \underbrace{ ch \big( V_{2n} \big) - e(f) \cdot ch \big( V_{2(n+n')} \big) } } \;+\; \underset{ \in \; H^{2(n + n')}\big( C_f; \, \mathbb{Q}/\mathbb{Z} \big) }{ \underbrace{ e(f) \cdot ch \big( V_{2(n+n')} \big) \,. } }

(See also Conner-Floyd 66, p. 100.)

The evaluation of this cohomology class on the fundamental class of C fC_f picks out the coefficient of ch(V 2(n+n))ch\big( V_{2(n+n')} \big), by (16):

C fch(V 2n)=e(f)/, \int_{C_f} ch \big( V_{2n} \big) \;=\; e(f) \;\;\; \in \; \mathbb{Q}/\mathbb{Z} \,,

and hence the claim follows.

Remark

The analogue statement of Prop. for the e e_{\mathbb{R}}-invariant (Def. ) may fail:

The e e_{\mathbb{R}}-invariant defined in terms of Adams operations (Def. ) is cc times the top degree coefficient of the Chern character on KOKUK \mathrm{O} \to K \mathrm{U} (the Pontrjagin character) with (Adams 66 (7.3)):

  1. c=1c = 1 for (n+n)=0mod8(n + n') = \,0\, mod \, 8 (here they coincide)

  2. c=12c = \tfrac{1}{2} for (n+n)=4mod8(n + n') = \,4\, mod \, 8 (here the e e_{\mathbb{R}}-invariant is finer).

Similarly (Adams 66 Prop. 7.14):

  • the e e_{\mathbb{R}}-invariant equals the e e_{\mathbb{C}}-invariant for n=0mod8n' = \,0\, mod \, 8,

  • but equals 12e \tfrac{1}{2}e_{\mathbb{C}} for n+n=4mod8n + n' = \,4\, mod \, 8 .

This means that the e e_{\mathbb{R}}-invariant is finer than the e e_{\mathbb{C}}-invariant.


As a cobordism invariant of U-manifolds with framed boundary

We discuss how the e-invariant in its Q/Z-incarnation (Def. ) has a natural formulation in cobordism theory (Conner-Floyd 66).

This is Prop. below; but first to recall some background:

Remark

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):

(17)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)

Remark

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

(18)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.

Proposition

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

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 (18) 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:

(19)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} \\ 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 (19) is the observation (Conner-Floyd 66, p. 100-101) that the representing map (17) for a (U,fr)(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


For E=KOE = KO

The Adams e e_{\mathbb{R}}-invariant in /\mathbb{Q}/\mathbb{Z}

Definition

Definition (via Adams operations) applies verbatim also with real (i.e. orthogonal) topological K-theory KO in place of complex topological K-theory KU. The resulting invariant is denoted e e_{\mathbb{R}} (Adams 66, p. 39).

But beware that e e_{\mathbb{R}} may fail to be equal to the coefficient of the top degree Chern character on KO (the Pontrjagin character), see Remark . That is, in fact, what makes e e_{\mathbb{R}} a finer invariant: It is either equal to e e_{\mathbb{C}} or to 12e \tfrac{1}{2}e_{\mathbb{C}}. (Adams 66 (7.3)).

As a cobordism invariant of SUSU-manifolds with framed boundary

An analogous but finer version of the cobordism-theoretic construction (above) 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 variant of (19):

(20)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).


Diagrammatic construction

under construction – not complete yet

While the Adams e-invariant of a map exists when the its d-invariant vanishes, the classical constructions above implicitly proceed via a finer invariant of maps equipped with a trivialization of their d-invariant. In order to bring this out more explicitly, the following is meant to be another, more abstractly homotopy theoretic, way to approach the construction of the e-invariant (following SS21). This perspective turns out to make various properties immediately manifest, notably the equality between Adams’s construction via the Chern character on KU and Conner-Floyd’s construction via the Todd character on MUFr.

  1. Preliminaries

  2. Unit cofiber cohomology theory

  3. The refined e-invariant

  4. Recovering Adams’s construction

  5. Recovering Conner-Floyd’s construction

Preliminaries

First some Notation:

  • For EE a spectrum and nn \in \mathbb{Z} we write

    • E nΩ Σ nEE^n \coloneqq \Omega^\infty \Sigma^n E for the nnth component space

    • E nπ n(E)E˜ 0(S n)π 0(E n)E_n \coloneqq \pi_n(E) \simeq \widetilde E{}^0(S^n) \simeq \pi_0(E^{-n}) for its homotopy groups.

  • For RR a ring we write HRH R for its Eilenberg-MacLane spectrum and H evRkΣ 2kHR H^{\mathrm{ev}}R \;\coloneqq\; \underset{k \in \mathbb{N}}{\oplus} \Sigma^{2k} H R for its connective even 2-periodic version.

The Setup is as above, which we recall for completeness:

  • Let n,dn, d \in \mathbb{N}, with d1d \geq 1,

  • consider a map S 2(n+d)1cS 2n S^{2(n+d)-1} \xrightarrow{\;c\;} S^{2n} representing a class in 𝕊 2d1\mathbb{S}_{2d-1}, which we will denote by the same name.

    Without restriction of generality we may assume that this class is non-trivial c0𝕊 2d1.c \neq 0 \,\in\, \mathbb{S}_{2d-1}\,.

  • We write C cC_c for the cofiber space of cc:

Unit cofiber cohomology theory

Definition

[Unit cofiber cohomology theory]

For EE a multiplicative cohomology theory (represented by a homotopy-commutative ring spectrum) with unit operation 𝕊e EE\mathbb{S} \overset{ e^E }{\longrightarrow} E we denote the corresponding homotopy cofiber-theory E/𝕊E/\mathbb{S}.

(In notation common around the Adams spectral sequence this would be “ΣE¯\Sigma \overline {E}” – as in Adams 74, theorem 15.1 page 319 – or just “E¯\overline{ E }” – as in Hopkins 99, Cor. 5.3).

Via the induced homotopy cofiber sequence

this comes with a canonical cohomology operation :E/𝕊Σ𝕊\partial \;\colon\; E/\mathbb{S} \longrightarrow \Sigma \mathbb{S} to shifted stable Cohomotopy:

Definition

[Induced cohomology operations on cofiber cohomology] Let EϕFE \overset{\phi}{\longrightarrow} F be a multiplicative cohomology operation, so that in particular it preserves the units, witnessed by a homotopy-commutative square on the left here:

Then passing to homotopy cofibers yields the induced cohomology operation ϕ/𝕊\phi/\mathbb{S} on cofiber theories (Def. ).

Example

[Chern character on cofiber of K-theory] The Chern character operation KUϕH ev\mathrm{KU} \overset{\phi}{\longrightarrow} H^{\mathrm{ev}}\mathbb{Q} is multiplicative, hence passes to an operation KU/𝕊ch/𝕊H ev\mathrm{KU}/\mathbb{S} \overset{ \;\mathrm{ch}/\mathbb{S}\; }{\longrightarrow} H^{\mathrm{ev}}\mathbb{Q} via Def. .

In a similar fashion we get:

Remark

There is a canonical splitting spl 0\mathrm{spl}_0 of the short exact sequence

namely that coming from the inclusion HH evH\mathbb{Q} \hookrightarrow H^{\mathrm{ev}}\mathbb{Q} (which is multiplicative, in particular preserves the unit):

where on the left we used our assumption that d0d \neq 0.

We will use this canonical splitting together with the canonical splitting of the even rational cohomology of a cofiber space S 2(n+d)1cS 2nC cS^{2(n+d)-1} \xrightarrow{ \;c\; } S^{2n} \xrightarrow{ \;\; } C_c given by the splitting into degrees 2(n+d)2(n+d) and 2n2n

which is induced in the same fashion from the inclusion HH evH\mathbb{Q} \hookrightarrow H^{\mathrm{ev}}\mathbb{Q}.

Both of these splittings are hence characterized by the fact that their corresponding retraction (see there) is projection onto rational cohomology in degree 2(n+d)2(n+d).

Lemma

(Cofiber EE-cohomology as extension of stable Cohomotopy by EE-cohomology)

For EE a multiplicative cohomology theory and XX a space, assume that the EE-Boardman homomorphism 𝕊˜ (X)β X E˜ (X)\widetilde {\mathbb{S}}{}^\bullet(X) \xrightarrow{ \;\beta^\bullet_X\; } \widetilde E^\bullet(X) is zero in degrees nn and n+1n + 1 – for instance in that XS k2X \simeq S^{k \geq 2}, n=0n = 0 and the groups E˜ 0(S k)=π k(E)\widetilde E^0(S^k) = \pi_k(E) have no torsion – then the cohomology operations i Ei^E, E\partial^E form a short exact sequence of cohomology groups:

(e.g. Stong 68, p. 102)

Proof

Generally, the long cofiber sequence of cohomology theories

induces a long exact sequence of cohomology groups (…)

Under the given assumption the two outermost morphisms shown are zero, and hence the sequence truncates as claimed.

The refined e-invariant

Proposition

Let EE be a multiplicative cohomology theory and n,dn,d \in \mathbb{N}. Consider the function that sends pairs consisting of

  1. the stable class [G n 𝕊(c)]\big[ G^{\mathbb{S}}_n (c) \big] of a map S n+d1cS nS^{n+d-1} \overset{\;c\;}{\longrightarrow} S^n

  2. the class of a trivialization H n1 E(c)H^E_{n-1}\!(c) of its d-invariant in EE-cohomology

to the class in (E/𝕊) d(E/\mathbb{S})_d of this diagram:

Then: This function respects the canonical fibrations of both sides over 𝕊 d1\mathbb{S}_{d-1}, i.e. it is a lift through the boundary map \partial.

See also this Prop. at d-invariant.

Proof

First, here is a quick formal argument to see that some such map does exist:

By Definition , an element in H n1Fluxes E(S n+d1)H_{n-1}Fluxes^E\big( S^{n+d-1} \big) is equivalently the class of a homotopy cone with tip Σ n+d1𝕊\Sigma^{n+d-1} \mathbb{S} over the cospan formed by the ring spectrum unit e Ee^E and the zero morphism:

But in Spectra homotopy cofiber sequences are homotopy fiber sequences (by this Prop.), so that by the universal property of homotopy fibers the class of the above diagram induces the class of a map
H n1 E(c):Σ n+d1𝕊fib(Σ ne E)Σ n1(E/𝕊\vdash {\color{orange} H^E_{n-1}(c) } \;\colon\; \Sigma^{n+d-1}\mathbb{S} \longrightarrow fib\big( \Sigma^{n} e^E \big) \,\simeq\, \Sigma^{n-1} (E/\mathbb{S}):

equipped with a homotopy from its image under \partial to cc.

This implies the claim, by

π 0Maps(Σ n+d1𝕊,Σ n1(E/𝕊))(E/𝕊) d. \pi_0 Maps \Big( \Sigma^{n+d - 1}\mathbb{S} \,,\, \Sigma^{n-1} (E/\mathbb{S}) \Big) \;\simeq\; (E/\mathbb{S})_{d} \,.


Second, to see that this map is realized as claimed (the following construction is close to the proof of Conner-Floyd 66, Theorem 16.2):

Let [S n+d1cS n]π n(S n+d1)\big[ S^{n + d - 1} \overset{c}{\longrightarrow} S^{n} \big] \;\in\; \pi^n\big(S^{n+d-1}\big) be a given class in Cohomotopy. We need to produce a map of the form

and show that it is a bijection onto this fiber, hence that the square is cartesian. To this end, we discuss the following homotopy pasting diagram, all of whose cells are homotopy cartesian:

For given H n1 E(c)H^E_{n-1}\!(c), this diagram is constructed as follows (where we say “square” for any single cell and “rectangle” for the pasting composite of any adjacent pair of them):

  • The two squares on the left are the stabilization of the homotopy pushout squares defining the cofiber space C cC_c and the suspension of S n+d1S^{n + d - 1}

  • The bottom left rectangle (with Σ n(e E)\Sigma^n(e^E) at its top) is the homotopy pushout defining Σ n(E/𝕊)\Sigma^n(E\!/\mathbb{S}).

  • The classifying map for the given (n1)(n-1)-flux, shown as a dashed arrow, completes a co-cone under the bottom left square. Thus the map M d{\color{magenta}M^d} forming the bottom middle square is uniquely implied by the homotopy pushout property of the bottom left square. Moreover, the pasting law implies that this bottom middle square is itself homotopy cartesian.

  • The bottom right square is the homotopy pushout defining \partial.

  • By the pasting law it follows that also the bottom right rectangle is homotopy cocartesian, hence that, after the two squares on the left, it exhibits the third step in the long homotopy cofiber sequence of Σ c\Sigma^\infty c. This means that its total bottom morphism is Σ +1c\Sigma^{\infty + 1} c, and hence that [M d]=[c]\partial \big[ M^d \big] = [c].

In conclusion, these construction steps yield a map H n1 E(c)M dH^E_{n-1}\!(c) \mapsto M^d over [Σ c][\Sigma^\infty c], as required.

Definition

The refined e-invariant e^ KU\widehat e_{KU} is the composite of

  1. the equivalence from Prop. for E=E = KU,

  2. the cofiber Chern character from Example ,

  3. the canonical splitting from Remark :

sending a trivialization of the d-invariant

to the class of this pasting composite:

Lemma

The values of the refined e-invariant from Def. for fixed Cohomotopy class [c][c] form an integer lattice inside the rational numbers:

e^ KU([c],[H 2d1 KU(c)])e^ KU([c],[H 2d1 KU(c)]) \widehat e_{\mathrm{KU}} \Big( [c], \big[ H^{\mathrm{KU}}_{2d-1}\!(c) \big] \Big) - \widehat e_{\mathrm{KU}} \Big( [c], \big[ {H}^\prime{}^{\mathrm{KU}}_{2d-1}\!(c) \big] \Big) \;\in\; \mathbb{Z}

Proof

Using the same three ingredients that enter Def. , we paste together the following commuting diagram, whose vertical middle composite is the refined e^ KU\widehat e_{KU}-invariant: