Contents

cohomology

# Contents

## Idea

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 $p_1^E$ (a generalized first Pontrjagin class) in a multiplicative Whitehead generalized cohomology theory $E$ is an extension of the 4-suspended ring unit in the cohomology ring $E_\bullet \,\simeq\, E^4\big( H\mathbb{P}^1\big)$ from the quaternionic projective space $\mathbb{H}P^1$ (the 4-sphere) to $H\mathbb{P}^\infty$:

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

(quaternionic $E$-orientation by extensions and their obstructions)

In terms of classifying maps, Def. means that a quaternionic orientation $p_1^E$ in $E$-cohomology theory is equivalently an extension (in the classical homotopy category) of the map $\Sigma^4 1 \,\colon\, \mathbb{H}P^1 \longrightarrow \Omega^{\infty -4} E$ (which classifies the suspended ring unit in the cohomology ring) along the canonical inclusion of quaternionic projective spaces

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

Notice that the quaternionic projective spaces form a cotower

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

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

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

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

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

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

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

of finite-stage extensions

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

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

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

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

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

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

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

Hence:

###### Proposition

If $E$ is a multiplicative Whitehead generalized cohomology theory whose graded cohomology ring $E_\bullet$ is trivial in degrees $4 n - 1$, then $E$ 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

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

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