nLab d-invariant

Idea

Let

(1)$$\big[ X \overset{f}{\longrightarrow} Y \big]$$

be a morphism in the stable homotopy category (for instance the stabilization of a morphism in the classical homotopy category) and let $E$ be a Whitehead-generalized cohomology theory (usually assumed to be at least multiplicative).

Then the $d$-invariant of $f$ with respect to $E$ (Adams 66, Sec. 3, p. 26, short for “degree invariant”, see Example below) is the class of the corresponding pullback morphism $f^\ast_E$, regarded as an element of the 0th Ext-group (i.e. the plain hom-object) between the $E$-cohomologies of $X$ and $Y$:

(2)$$d(f) \;\coloneqq\; f^\ast_E \;\in\; \big( E^\bullet(Y); \, E^\bullet(X) \big) \;=\; Ext^0 \big( E^\bullet(Y); \, E^\bullet(X) \big) \,.$$

The fancy name is justified by the fact that this is beginning of a hierarchy of more interesting invariants of stable maps seen in $E$-cohomology, each of which defined when the previous one vanishes: The next is the e-invariant, then comes the f-invariant (e.g. BL 09, Section 2). These are the elements that appear in the first lines on the second page of the $E$-Adams spectral sequence for $[X,Y]_\bullet$.

Properties

Relation to the Hopf degree

Example

(d-invariant specializes to degree of a continuous function)

In the special case that

• $E = H\mathbb{Z}$ is integral ordinary cohomology;

• $X$ and $Y$ are connected closed oriented topological manifolds of the same dimension $n = dim(X) = dim(Y)$

then the $n$-component of the $d$-invariant (2) is the degree of $f$

(3)$$d^n(f) = deg(f) \;\in\; Hom_{Ab} \big( H^n(Y;\mathbb{Z}), \, H^n(YX;\mathbb{Z}) \big) \;\simeq\; \mathbb{Z} \,.$$

If moreover $Y = S^n$ is the n-sphere, so that $f$ represents a class in the $n$th Cohomotopy of $X$, then $\widetilde H^\bullet(Y; \mathbb{Z})$ is concentrated on $\mathbb{Z}$ in degree $n$ and hence the $n$ component $d^n(f)$ is the full information in the d-invariant.

In this case, the Hopf degree theorem says that the degree of $f$, and hence the d-invariant of $f$ in integral cohomology, completely characterizes the homotopy class $[f]$.

Trivializations of the d-invariant

When the d-invariant vanishes, then the e-invariant exists. But in fact, classical constructions of the e-invariant proceed via a choice of a trivialization of the d-invariant and conclude by showing the result to be independent of this choice (this is meant to be brought out at Adams e-invariant – Construction via unit cofiber theories).

Here we discuss the space of choices of trivializations of the d-invariant (following SS 21):

Definition

[Set of trivializations of the d-invariant] For $E$ a multiplicative cohomology theory and $n, d \in \mathbb{N}$, write

(4)$$\begin{aligned} H^E_{n-1}\!Fluxes\!\big( S^{n+d-1} \big) & \;\coloneqq\; \underset{ [c] \in \mathbb{S}_{d-1} }{\sqcup} \pi_0 \mathrm{Paths}_0^{c^\ast (1^E)} \Big( \mathrm{Maps}^{\ast/}\! \big( S^{n+d-1} \,,\, E^{n} \big) \Big) \end{aligned}$$

for the set of tuples consisting of the stable Cohomotopy class $\big[ G^{\mathbb{S}}_{n}\!(c) \big]$ of a map $S^{n+d-1} \xrightarrow{\;c\;}S^{n}$ and the 2-homotopy class $\big[ H^E_{n-1}\!(c) \big]$ of a trivialization (if any) of its d-invariant $c^\ast(1^E)$ in $E$-cohomology.

So an element of $H^E_{n-1}\!Fluxes\!\big( S^{n+d-1} \big)$ (4) is the 2-homotopy class relative the boundary of a homotopy coherent diagramof the following form:

Definition

[Unit cofiber cohomology theory]

For $E$ a multiplicative cohomology theory with unit map $\mathbb{S} \overset{ e^E }{\longrightarrow} E$ we denote the corresponding homotopy cofiber-theory as follows:

Proposition

[Trivializations of d-invariant are classes in cofiber theory] There is a map between the set (4) of trivializations of the d-invariant and the cohomology group of the unit cofiber theory $E\!/\mathbb{S}$ (Def. ):

compatible with the fibrations of both over the underlying stable Cohomotopy classes $G^{\mathbb{S}}_n Fluxes \big( S^{n+d-1} \big) \;\coloneqq\; \widetilde {\mathbb{S}}{}^d \big( S^{n + d - 1}\big) \,\simeq\, \mathbb{S}_{d-1}$.

We give two proofs: A quick abstract one and a more explicit one that proceeds via classes on the cofiber space of $c$. The latter is close to the old argument of Conner-Floyd 66, Thm. 16.2 (there for $E/\mathbb{S} =$ MUFr, see this section for more).

Proof

[quick abstract]

By Definition , an element in $H_{n-1}Fluxes^E\big( S^{n+d-1} \big)$ is equivalently the class of a homotopy cone with tip $\Sigma^{n+d-1} \mathbb{S}$ over the cospan formed by the ring spectrum unit $e^E$ 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 is equivalently the class of a map from $\Sigma^{n+d-1}\mathbb{S}$ to $fib\big( \Sigma^{n} e^E \big) \,\simeq\, \Sigma^{n-1} (E/\mathbb{S}$):

This implies the claim, by

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

Proof

[more explicit, Conner-Floyd-style]

Let $\big[ S^{n + d - 1} \overset{c}{\longrightarrow} S^{n} \big] \;\in\; \pi^n\big(S^{n+d-1}\big)$ 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 $H^E_{n-1}\!(c)$, 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 $C_c$ and the suspension of $S^{n + d - 1}$

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

• The classifying map for the given $(n-1)$-flux, shown as a dashed arrow, completes a co-cone under the bottom left square. Thus the map ${\color{magenta}M^d}$ 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 cartesian, hence that, after the two squares on the left, it exhibits the third step in the long homotopy cofiber sequence of $\Sigma^\infty c$. This means that its total bottom morphism is $\Sigma^{\infty + 1} c$, and hence that $\partial \big[ M^d \big] = [c]$.

In conclusion, these construction steps yield a map map $H^E_{n-1}\!(c) \mapsto M^d$ which is as required.

Related concepts

• degree of a continuous function, Hopf degree theorem

• e-invariant

• f-invariant

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution

References

The notion was introduced, together with that of the e-invariant, in:

• John Adams, Section 3 of: On the groups $J(X)$ IV, Topology 5: 21 (1966) (pdf, doi:10.1016/0040-9383(66)90004-8)

Discussion in the broader context including also the f-invariant:

• Mark Behrens, Gerd Laures, $\beta$-Family congruences and the $f$-invariant, Geometry & Topology Monographs 16 (2009) 9–29 (arXiv:0809.1125, doi: 10.2140/gtm.2009.16.9)