nLab
Boardman homomorphism

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Stable Homotopy theory

Cohomology

cohomology

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Idea

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

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

For E=HE = H \mathbb{Z} the Eilenberg-MacLane spectrum for ordinary homology, then this reduces to the Hurewicz homomorphism π ()H ()\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 EE:

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

Unifying these two cases, there is the bivariant Boardman homomorphism

[X,Y] [X,Y𝕊] [X,YE] . [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 grous of MU via the homology of MU (Quillen's theorem on MU), see below.

Examples

From stable cohomotopy to ordinary cohomology

Consider the unit morphism

𝕊H \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:π n(X)H n(X,) b^n \;\colon\; \pi^n(X) \longrightarrow H^n(X, \mathbb{Z})

from the stable cohomotopy of XX in degree nn to its ordinary cohomology in degree nn.

Proposition

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

If XX is a CW-spectrum which

  1. is (m-1)-connected

  2. of dimension dd \in \mathbb{N}

then

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

    mnd1 m \leq n\leq d -1

    is a ρ¯ dn\overline{\rho}_{d-n}-torsion group:

    ρ¯ dnker(b n)0 \overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0
  2. the cokernel of the Boardman homomorphism b nb^n (1) for

    mnd2 m \leq n \leq d - 2

    is a ρ¯ dn1\overline{\rho}_{d-n-1}-torsion group:

    (2)ρ¯ dn1coker(b n)0 \overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0

where

ρ¯ i{1 | i1 j=1iexp(π j(𝕊)) | otherwise \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 i\leq i.

(Arlettaz 04, theorem 1.2)

Example

(estimates for torsion of cokernel of Boardman homomorphism)

Let XX be a manifold

then Prop. asserts that the cokernel of the Boardman homomorphism

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

in

  • degree n=4n = 4

is 2-torsion:

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

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

ρ¯ dn1 =ρ¯ 1 =exp(π 1(𝕊)) =exp(/2) =2. \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=7d = 7 but sticking to degree n=4n = 4, then the estimate is that the cokernel is 4-torsion,

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

since then

ρ¯ dn1 =ρ¯ 2 =exp(π 1(𝕊))exp(π 2(𝕊)) =exp(/2)exp(/2) =22 =4. \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=8d = 8 we

ρ¯ dn1 =ρ¯ 3 =exp(π 1(𝕊))exp(π 2(𝕊))exp(π 3(𝕊)) =exp(/2)exp(/2)exp(/24) =226 =24. \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} \,.

For complex oriented cohomology theories

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)

Examples

Proposition

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

Consider the binary icosahedral group 2I2 I and its action on the 7-sphere induced via the identification S 7S(×)S^7 \simeq S(\mathbb{H} \times \mathbb{H}) from the diagonal of the canonical action of 2I2I on the quaternions \mathbb{H} induced via it being a finite subgroup of SU(2).

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

𝕊 4(S 7/2I)βH 4(S 7/2I,) \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(S 7/2I,π s 0=π 0 s=)d 3H 6(S 7/2I,π s 1=π 1 s=/2) 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(S 7/2I,π s 0=π 0 s=)d 3H 7(S 7/2I,π s 2=π 2 s=/2) 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(,π s 0)H^4(-,\pi^0_s) vanish simply for dimensional reasons as S 7S^7 is of dimension 7, while there are no differentials into H 4(,π s 0)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 2d_2 to vanish, it is sufficient that

H 6(S 7/2I,π s 1=π 1 s=/2)0 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

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

we have

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

where B(2I)*(2I)B (2 I) \simeq \ast \sslash (2I) is the classifying space of 2I2I (see e.g. at infinity-action).

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

0Ext 1(H 5(B(2I),),/2)H 6(B(2I),/2)Hom Ab(H 6(B(2I),),/2)0. 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(B(2I),)0AAAH 6(B(2I),)0 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 ADESU(2)G_{ADE} \subset SU(2) we have (by this Prop.)

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

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

This shows that d 2d_2 vanishes on H 4(,π 0)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 nSq^n (e.g. here, but let’s add a more canonical reference).

This means first of all that if d 2d_2 is not trivial then d 2=Sq 2d_2 = Sq^2. But since that vanishes on H 4(,π 0)H^4(-,\pi^0) by the above argument, and on H 7(,π 2)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 3d_3 = Sq^3.

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

References

Named after Michael Boardman.

Last revised on February 17, 2019 at 06:09:03. See the history of this page for a list of all contributions to it.