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quaternionic oriented cohomology theory

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cohomology

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Quaternionic oriented cohomology theory is the analog of complex oriented cohomology theory as complex vector bundles are replaced by quaternionic vector bundles:

Definition

Definition

(quaternionic oriented cohomology theory)

A quaternionic orientation p 1 Ep_1^E (a generalized first Pontrjagin class) in a multiplicative Whitehead generalized cohomology theory EE is an extension of the 4-suspended ring unit in the cohomology ring E E 4(H 1)E_\bullet \,\simeq\, E^4\big( H\mathbb{P}^1\big) from the quaternionic projective space P 1\mathbb{H}P^1 (the 4-sphere) to H H\mathbb{P}^\infty:

(1)P 1 Σ 41 Ω 4E p 1 E H \array{ \mathbb{H}P^1 &\overset{\Sigma^4 1}{\longrightarrow}& \Omega^{\infty - 4} E \\ \big\downarrow & \nearrow_{ \mathrlap{p_1^E} } \\ H \mathbb{P}^\infty }
Remark

(quaternionic EE-orientation by extensions and their obstructions)

In terms of classifying maps, Def. means that a quaternionic orientation p 1 Ep_1^E in EE-cohomology theory is equivalently an extension (in the classical homotopy category) of the map Σ 41:P 1Ω 4E\Sigma^4 1 \,\colon\, \mathbb{H}P^1 \longrightarrow \Omega^{\infty -4} E (which classifies the suspended ring unit in the cohomology ring) along the canonical inclusion of quaternionic projective spaces

(2)P 1 Σ 41 E Ω 4E p 1 E P . \array{ \mathbb{H}P^1 & \overset{ \Sigma^4 1_E }{ \longrightarrow } & \Omega^{\infty - 4} E \\ \big\downarrow & \nearrow \mathrlap{ {}_{p_1^E} } \\ \mathbb{H}P^\infty \,. }

Notice that the quaternionic projective spaces form a cotower

*=P 0P 1P 2P 3P =limP \ast \,=\, \mathbb{H}P^0 \hookrightarrow \mathbb{H}P^1 \hookrightarrow \mathbb{H}P^2 \hookrightarrow \mathbb{H}P^3 \hookrightarrow \cdots \hookrightarrow \mathbb{H}P^\infty \,=\, \underset{\longrightarrow}{\lim} \mathbb{H}P^\bullet

where each inclusion stage is (by this Prop., see at cell structure of projective spaces) the coprojection of a pushout of topological spaces (or rather: of pointed topological spaces) of the form

D 4n+4 P n+1 (po) S 4n+3 h 4n+3 P n \array{ D^{4n+4} & \overset{}{\longrightarrow} & \mathbb{H}P^{n+1} \\ \big\uparrow &\mathclap{^{_{(po)}}}& \big\uparrow \\ S^{4n+3} &\underset{h^{4n+3}_{\mathbb{H}}}{\longrightarrow}& \mathbb{H}P^n }

(where h 4n+3h^{4n+3}_{\mathbb{H}} is the quaternionic Hopf fibration in dimension 4n+34n+3) hence of a homotopy pushout of underlying homotopy types (rather: of pointed homotopy types) of this form:

* P n+1 (hpo) S 4n+3 h 4n+3 P n \array{ \ast & \overset{}{\longrightarrow} & \mathbb{H}P^{n+1} \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow \\ S^{4n+3} &\underset{h^{4n+3}_{\mathbb{H}}}{\longrightarrow}& \mathbb{H}P^n }

Therefore, a quaternionic orientation by extension (1) is equivalently the homotopy colimiting map of a sequence

(Σ 41=p 1 E,0,p 1 E,1,p 1 E,2,) \big( \Sigma^4 1 \,=\, p_1^{E,0} ,\, p_1^{E,1} ,\, p_1^{E,2} ,\, \cdots \big)

of finite-stage extensions

* P n+1 p 1 E,n+1 Ω 4E (hpo) p 1 E,n S 4n+3 h 4n+3 P n. \array{ \ast & \overset{}{\longrightarrow} & \mathbb{H}P^{n+1} & \overset{ p_1^{E,n+1} }{\longrightarrow} & \Omega^{\infty - 4} E \\ \big\uparrow &\mathclap{^{_{(hpo)}}}& \big\uparrow & \nearrow \mathrlap{ {}_{p_1^{E,n}} } \\ S^{4n+3} &\underset{h^{4n+3}_{\mathbb{H}}}{\longrightarrow}& \mathbb{H}P^n \,. }

Moreover, by the defining universal property of the homotopy pushout, the extension p 1 E,n+1p_1^{E,n+1} of p 1 E,np_1^{E,n} is equivalently a choice of homotopy which trivializes the pullback of p 1 E,np_1^{E,n} to the 4n+3-sphere:

* Ω 4E p 1 E,n+1 p 1 E,n S 4n+3 h 4n+3 P n. \array{ \ast & \overset{}{\longrightarrow} & \Omega^{\infty - 4} E \\ \big\uparrow & {}_{ p_1^{E,n+1} } \seArrow & \big\uparrow \mathrlap{ ^{_{ p_1^{E,n} }} } \\ S^{4n+3} &\underset{ h^{4n+3}_{\mathbb{H}} }{\longrightarrow}& \mathbb{H}P^n \,. }

This means, first of all, that the non-triviality of the pullback class

(h 4n+3) *(p 1 E,n)E˜ 4(S 4n+3)E 4n1 \big( h^{4n+3}_{\mathbb{C}} \big)^\ast ( p_1^{E,n} ) \;\in\; \widetilde E^4 \big( S^{4n+3} \big) \;\simeq\; E_{4n - 1}

is the obstruction to the existence of the extension/orientation at this stage.

It follows that if these obstructions all vanish, then a quaternionic EE-orientation does exist. A sufficient condition for this is, evidently, that the reduced EE-cohomology of all (4n1)(4n-1)-dimensional spheres vanishes.

Hence:

Proposition

If EE is a multiplicative Whitehead generalized cohomology theory whose graded cohomology ring E E_\bullet is trivial in degrees 4n14 n - 1 , then EE admits a quaternionic orientation (Def. ).

Examples

  • ordinary cohomology has coefficient ring concentrated in degree 0, and hence satisfies the sufficient condition of Prop. to admit quaternionic orientation;

  • KO-theory (orthogonal topological K-theory) has coefficient ring

    KO n{ | n=0mod8 2 | n=1mod8 2 | n=2mod8 0 | n=3mod8 | n=4mod8 0 | n=5mod8 0 | n=6mod8 0 | n=7mod8, KO_n \,\simeq\, \left\{ \array{ \mathbb{Z} &\vert& n = 0 \,mod\, 8 \\ \mathbb{Z}_2 &\vert& n = 1 \,mod\, 8 \\ \mathbb{Z}_2 &\vert& n = 2 \,mod\, 8 \\ 0 &\vert& n = 3 \,mod\, 8 \\ \mathbb{Z} &\vert& n = 4 \,mod\, 8 \\ 0 &\vert& n = 5 \,mod\, 8 \\ 0 &\vert& n = 6 \,mod\, 8 \\ 0 &\vert& n = 7 \,mod\, 8 } \right. \,,

    and hence satisfies the sufficient condition of Prop. to admit quaternionic orientation;

    (see also Laughton 08, Example 2.2.7)

  • Among quaternionic oriented cohomology theories, quaternionic cobordism is universal in the fashion analogous to the universal complex orientation on MU (Laughton 08, Example 2.2.9)

References

Last revised on February 16, 2021 at 09:16:32. See the history of this page for a list of all contributions to it.