nLab Boardman homomorphism

Idea

Given any homotopy commutative ring spectrum $(E, \mu, e)$, then the Boardman homomorphism is the homomorphism from stable homotopy groups (hence from stable homotopy homology theory) to $E$-generalized homology groups that is induced by smash product with the unit map $e \colon \mathbb{S} \longrightarrow E$ from the sphere spectrum:

$$\pi_\bullet(-) \simeq \pi_\bullet(\mathbb{S} \wedge (-)) \longrightarrow \pi_\bullet(E \wedge (-)) = E_\bullet(-) \,.$$

For $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum for ordinary homology, then this reduces to the Hurewicz homomorphism $\pi_\bullet(-) \to H_\bullet(-)$.

Dually, there is the Boardman homomorphism from stable cohomotopy to generalized cohomology induced under forming mapping spectra into the unit map of $E$:

$$\pi^\bullet(-) \simeq \pi_\bullet([(-),\mathbb{S}]) \longrightarrow \pi_\bullet([(-),E]) = E^\bullet(-) \,.$$

Unifying these two cases, there is the bivariant Boardman homomorphism (Adams 74, p. 58)

$$[X, Y]_\bullet \;\simeq\; [X, Y \wedge \mathbb{S}]_\bullet \longrightarrow [X,Y \wedge E]_\bullet \,.$$

Since generalized homology/generalized cohomology is typically more tractable than homotopy groups/cohomotopy (in particular when homology spectra split), the Boardman homomorphism is often used to partially reduce computations of the latter in terms of computations of the former.

One example is the computation of the homotopy groups of MU via the homology of MU (Quillen's theorem on MU), see below.

Examples

In ordinary cohomology

Consider the unit morphism

$$\mathbb{S} \longrightarrow H \mathbb{Z}$$

from the sphere spectrum to the Eilenberg-MacLane spectrum of the integers. For any topological space/spectrum postcomposition with this morphism induces Boardman homomorphisms of cohomology groups (in fact of commutative rings)

(1)$$b^n \;\colon\; \pi^n(X) \longrightarrow H^n(X, \mathbb{Z})$$

from the stable cohomotopy of $X$ in degree $n$ to its ordinary cohomology in degree $n$.

Proposition

(bounds on (co-)kernel of Boardman homomorphism from stable cohomotopy to integral cohomology)

If $X$ is a CW-spectrum which

1. is (m-1)-connected

2. of dimension $d \in \mathbb{N}$

then

1. the kernel of the Boardman homomorphism $b^n$ (1) for

   $$m \leq n\leq d -1$$

   is a $\overline{\rho}_{d-n}$-torsion group:

   $$\overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0$$

2. the cokernel of the Boardman homomorphism $b^n$ (1) for

   $$m \leq n \leq d - 2$$

   is a $\overline{\rho}_{d-n-1}$-torsion group:

   (2)$$\overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0$$

where

$$\overline{\rho}_{i} \;\coloneqq\; \left\{ \array{ 1 &\vert& i\leq 1 \\ \underoverset{j = 1}{i}{\prod} exp\left( \pi_j\left( \mathbb{S}\right) \right) &\vert& \text{otherwise} } \right.$$

is the product of the exponents of the stable homotopy groups of spheres in positive degree $\leq i$.

(Arlettaz 04, theorem 1.2)

Example

(estimates for torsion of cokernel of Boardman homomorphism)

Let $X$ be a manifold

• of dimension $d = 6$

• simply connected with $\pi_2(X) \neq 0$

then Prop. asserts that the cokernel of the Boardman homomorphism

$$\beta^4 \;\colon\; \mathbb{S}^4(X) \longrightarrow H^4( X, \mathbb{Z} )$$

in

• degree $n = 4$

is 2-torsion:

$$2 coker(\beta^4) \;=\; 0 \,.$$

This is because in this case (2) gives that the relevant torsion degree is

$$\begin{aligned} \overline{\rho}_{d-n-1} & = \overline{\rho}_{1} \\ & = \exp( \pi_1(\mathbb{S}) ) \\ & = \exp( \mathbb{Z}/2 ) \\ & = 2 \end{aligned} \,.$$

Similarly, if instead the manifold has dimension $d = 7$ but sticking to degree $n = 4$, then the estimate is that the cokernel is 4-torsion,

$$4 coker(\beta^4) \;=\; 0 \,.$$

since then

$$\begin{aligned} \overline{\rho}_{d-n-1} & = \overline{\rho}_{2} \\ & = \exp( \pi_1(\mathbb{S}) ) \cdot \exp( \pi_2(\mathbb{S}) ) \\ & = \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/2 ) \\ & = 2 \cdot 2 \\ & = 4 \end{aligned} \,.$$

Next for $d = 8$ we

$$\begin{aligned} \overline{\rho}_{d-n-1} & = \overline{\rho}_{3} \\ & = \exp( \pi_1(\mathbb{S}) ) \cdot \exp( \pi_2(\mathbb{S}) ) \cdot \exp( \pi_3(\mathbb{S}) ) \\ & = \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/2 ) \cdot \exp( \mathbb{Z}/{24} ) \\ & = 2 \cdot 2 \cdot 6 \\ & = 24 \end{aligned} \,.$$

Proposition

(Boardman isomorphism on 2-sphere mod binary icosahedral group)

Consider the binary icosahedral group $2 I$ and its action on the 7-sphere induced via the identification $S^7 \simeq S(\mathbb{H} \times \mathbb{H})$ from the diagonal of the canonical action of $2I$ on the quaternions $\mathbb{H}$ induced via it being a finite subgroup of SU(2).

On the quotient space $S^7/2 I$ the Boardman homomorphism in degree 4 is an isomorphism

$$\mathbb{S}^4\left( S^7/2I \right) \underoverset{\simeq}{\beta}{\longrightarrow} H^4\left( S^7/2I , \mathbb{Z} \right)$$

from stable cohomotopy in degree 4 to integral cohomology in degree 4.

Proof

In terms of the Atiyah-Hirzebruch spectral sequence for stable cohomotopy it is sufficient to see that the two differentials

$$H^4\left( S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z} \right) \overset{d_3}{\longrightarrow} H^6\left( S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2 \right)$$

and

$$H^4\left( S^7/2I, \pi^0_s = \pi^s_0 =\mathbb{Z} \right) \overset{d_3}{\longrightarrow} H^7\left( S^7/2I, \pi^{-2}_s = \pi^s_{2} =\mathbb{Z}/2 \right)$$

both vanish (all higher differentials on $H^4(-,\pi^0_s)$ vanish simply for dimensional reasons as $S^7$ is of dimension 7, while there are no differentials into $H^4(-,\pi^0_s)$ simply because the sphere spectrum is connective, so that the stable homotopy groups of spheres vanish in negative degree).

For $d_2$ to vanish, it is sufficient that

$$H^6\left( S^7/2I, \pi^{-1}_s = \pi^s_{1} =\mathbb{Z}/2 \right) \;\simeq\; 0$$

We now first show that this is the case:

First, by the Gysin sequence for the spherical fibration

$$\array{ S^7 &\longrightarrow& S^7/SI \\ && \downarrow \\ && B (2 I) }$$

we have

$$H^6\left( S^7/2I, \, \mathbb{Z}/2 \right) \;\simeq\; H^6\left( B(2I),\, \mathbb{Z}/2 \right) \,,$$

where $B (2 I) \simeq \ast \sslash (2I)$ is the classifying space of $2I$ (see e.g. at infinity-action).

Moreover, by the universal coefficient theorem (this Prop.) we have a short exact sequence

$$0 \to Ext^1(H_{5}\big(B(2I), \mathbb{Z}), \mathbb{Z}/2\big) \longrightarrow H^6\big(B(2I), \mathbb{Z}/2\big) \longrightarrow Hom_{Ab}\big( H_6( B(2I), \mathbb{Z}) , \mathbb{Z}/2 \big) \to 0 \,.$$

This means that it is sufficient to see that

$$H_{5}\big(B(2I), \mathbb{Z}) \simeq 0 \phantom{AAA} H_{6}\big(B(2I), \mathbb{Z}) \simeq 0$$

But for every finite subgroup of SU(2) $G_{ADE} \subset SU(2)$ we have (by this Prop.)

$$H_{5}\big(B(2I), \mathbb{Z}) \simeq G^{ab}_{ADE} \phantom{AAA} H_{6}\big(B(2I), \mathbb{Z}) \simeq 0$$

where $G^{ab}_{ADE}$ is the abelianization of $G_{ADE}$. Specifically for $G_{ADE} = 2I$ this does vanish: the binary icosahedral group is a perfect group (this Prop.).

This shows that $d_2$ vanishes on $H^4(-, \pi^0)$.

Now by a standard argument, the AHSS-differentials between ordinary cohomology groups are stable cohomology operations, and thus, if non-trivial, must be the Steenrod operations $Sq^n$ (e.g. here, but let’s add a more canonical reference).

This means first of all that if $d_2$ is not trivial then $d_2 = Sq^2$. But since that vanishes on $H^4(-,\pi^0)$ by the above argument, and on $H^7(-,\pi^2)$ for dimension reasons, so that the relevant entries pass as ordinary cohomology groups to the third page of the spectral sequence, it follows similarly that $d_3 = Sq^3$.

But by the Adem relation $Sq^3 = Sq^1 \circ Sq^2$, the vanishing of $Sq_2$ on $H^4(-,\pi^0)$ then also implies the vanishing of $d_3$ on this entry.

In complex oriented cohomology

Used for complex oriented cohomology theories and proof of Quillen's theorem on MU via the homology of MU (…)

(Adams 74, pages 60-62, Lurie 10, lecture 7)

In topological modular forms

See Boardman homomorphism in tmf

Related concepts

• Hopf degree theorem

References

According to Hunton 95, the concept was introduced, in print, in:

• John Michael Boardman, The eightfold way to BP-operations, in: Current trends in algebraic topology, pp. 187–226, Canadian Mathematical Society Proceedings, 2, Part 1. Providence 1982 (ISBN:978-0-8218-6003-8)

Further discussion:

• John Frank Adams, part II, section 6 of: Stable homotopy and generalised homology (1974)

• John Hunton, The Boardman homomorphism, Contemporary Mathematics 181, 251-251, 1995 (GoogleBooks)

• Stanley Kochmann, section 4.3 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Dominique Arlettaz, The generalized Boardman homomorphisms, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 (doi:10.2478/BF02475949)

• Jacob Lurie, lecture 7 of Chromatic Homotopy Theory, 2010, (pdf)

• Akhil Mathew, Torsion exponents in stable homotopy and the Hurewicz homomorphism, Algebr. Geom. Topol. 16 (2016) 1025-1041 (arXiv:1501.07561)

• Hadi Zare, On the image of the unstable Boardman map (arXiv:1806.07079)

On the Boardman homomorphism (generalized Hurewicz homomorphism) to tmf:

• Mark Behrens, Mark Mahowald, J.D. Quigley, The 2-primary Hurewicz image of $tmf$ (arXiv:2011.08956)